A labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x, y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). We say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labeled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges, every label either occurs exactly \( k \) times or does not occur at all. In this paper, we prove that Butterfly and Benes networks are minimally \( 2^r \)-equitable, where \( r \) is the dimension of the networks.
The aim of this article is focused on developing an efficient algorithm for simulating Cellular Neural Network arrays (CNNs) using numerical integration techniques. The role of the simulator is that it is capable of performing raster simulation for any kind as well as any size of input image. It is a powerful tool for researchers to investigate the potential applications of CNN. This article proposes an efficient pseudo code for exploiting the latency properties of Cellular Neural Networks along with well known numerical integration algorithms. Simulation results and comparison have also been presented to show the efficiency of the numerical integration algorithms. It is observed that the Runge-Kutta (RK) sixth order algorithm outperforms well in comparison with the Explicit Euler, RK-Gill and RK-fifth order algorithms.
Image compression is the key technology in the development of various multimedia applications. Vector quantization is a universal and powerful technique to compress a data sequence, such as speech or image, resulting in some loss of information. In VQ, minimization of Mean Square Error (MSE) between code book vectors and training vectors is a non-linear problem. Traditional LBG types of algorithms used for designing the codebooks for Vector Quantizer converge to a local minimum, which depends on the initial code book. Memetic algorithms (MAs) are population-based meta-heuristic search approaches that have been receiving increasing attention in the recent years. These algorithms are inspired by models of natural systems that combine the evolutionary adaptation of a population with individual learning within the lifetimes of its members. It has shown to be successful and popular for solving optimization problems. In this paper, we present a new approach to vector quantization based on memetic algorithm. Simulations indicate that vector quantization based on memetic algorithm has better performance in designing the optimal codebook for Vector Quantizer than conventional LBG algorithm. The Peak Signal to Noise Ratio (PSNR) is used as an objective measure of reconstructed image quality.
Let \( G = (V, E) \) be a connected simple graph. Let \( u, v \in V(G) \). The detour distance, \( D(u, v) \), between \( u \) and \( v \) is the distance of a longest path from \( u \) to \( v \). E. Sampathkumar defined the detour graph of \( G \), denoted by \( D(G) \), as follows: \( D(G) \) is an edge-labelled complete graph on \( n \) vertices, where \( n = |V(G)| \), the edge label for \( uv \), \( u, v \in V(K_n) \), being \( D(u, v) \). Any edge-labelled complete graph need not be the detour graph of a graph. In this paper, we characterize detour graphs of a tree. We also characterize graphs for which the detour distance sequences are given.
Let \( M = \{v_1, v_2, \ldots, v_n\} \) be an ordered set of vertices in a graph \( G \). Then \( (d(u,v_1), d(u,v_2), \ldots, d(u,v_n)) \) is called the \( M \)-coordinates of a vertex \( u \) of \( G \). The set \( M \) is called a metric basis if the vertices of \( G \) have distinct \( M \)-coordinates. A minimum metric basis is a set \( M \) with minimum cardinality. The cardinality of a minimum metric basis of \( G \) is called minimum metric dimension. This concept has wide applications in motion planning and in the field of robotics. In this paper we provide bounds for minimum metric dimension of certain classes of enhanced hypercube networks.
We study the spectral radius of graphs with \(n\) vertices and a \(k\)-vertex cut and describe the graph which has the maximal spectral radius in this class. We also discuss the limit point of the maximal spectral radius.
Consider lattice paths in \(\mathbb{Z}^2\) taking unit steps north (N) and east (E). Fix positive integers \(r,s\) and put an equivalence relation on points of \(\mathbb{Z}^2\) by letting \(v,w\) be equivalent if \(v-w = \ell(r,s)\) for some \(k \in \mathbb{Z}\). Call a lattice path \({valid}\) if whenever it enters a point \(v\) with an E-step, then any further points of the path in the equivalence class of \(v\) are also entered with an E-step. Loehr and Warrington conjectured that the number of valid paths from \((0,0)\) to \((nr,ns)\) is \({\binom{r+s}{nr}}^n\). We prove this conjecture when \(s=2\).
Given integers \(m \geq 2, r \geq 2\), let \(q_m(n), q_0^{(m)}(n), b_r^{(m)}(n)\) denote respectively the number of \(m\)-colored partitions of \(n\) into: distinct parts, distinct odd parts, and parts not divisible by \(r\).We obtain recurrences for each of the above-mentioned types of partition functions.
A reflection of a regular map on a Riemann surface fixes some simple closed curves, which are called \({mirrors}\). Each mirror passes through some of the geometric points (vertices, face-centers and edge-centers) of the map such that these points form a periodic sequence which we call the \({pattern}\) of the mirror. For every mirror there exist two particular conformal automorphisms of the map that fix the mirror setwise and rotate it in opposite directions. We call these automorphisms the \({rotary\; automorphisms}\) of the mirror. In this paper, we first introduce the notion of pattern and then describe the patterns of mirrors on surfaces. We also determine the rotary automorphisms of mirrors. Finally, we give some necessary conditions under which all reflections of a regular map are conjugate.
We prove the non-existence of maximal partial spreads of size \(76\) in \(\text{PG}(3,9)\). Relying on the classification of the minimal blocking sets of size 15 in \(\text{PG}(2,9)\) \([22]\), we show that there are only two possibilities for the set of holes of such a maximal partial spread. The weight argument of Blokhuis and Metsch \([3]\) then shows that these sets cannot be the set of holes of a maximal partial spread of size \(76\). In \([17]\), the non-existence of maximal partial spreads of size \(75\) in \(\text{PG}(3,9)\) is proven. This altogether proves that the largest maximal partial spreads, different from a spread, in \(\text{PG}(3,q = 9)\) have size \(q^2 – q + 2 = 74\).
A weakly connected dominating set \(W\) of a graph \(G\) is a dominating set such that the subgraph consisting of \(V(G)\) and all edges incident on vertices in \(W\) is connected. In this paper, we generalize it to \([r, R]\)-dominating set which means a distance \(r\)-dominating set that can be connected by adding paths with length within \(R\). We present an algorithm for finding \([r, R]\)-dominating set with performance ratio not exceeding \(ln \Delta_r + \lceil \frac{2r+1}{R}\rceil – 1\), where \(\Delta_r\) is the maximum number of vertices that are at distance at most \(r\) from a vertex in the graph. The bound for size of minimum \([r, R]\)-dominating set is also obtained.
For \(n \in \mathbb{N}\), let \(a_n\) count the number of ternary strings of length \(n\) that contain no consecutive \(1\)s. We find that \(a_n = \left(\frac{1}{2}+\frac{\sqrt{3}}{3}\right)\left(1 + \sqrt{3}\right)^n – \left(\frac{1}{2}-\frac{\sqrt{3}}{3}\right)\left(1 – \sqrt{3}\right)^n\). For a given \(n \geq 0\), we then determine the following for these \(a_n\) ternary strings:
(1)the number of \(0’\)s, \(1’\)s, and \(2’\)s;(2)the number of runs;(3) the number of rises, levels, and descents; and
(4)the sum obtained when these strings are considered as base \(3\) integers.
Following this, we consider the special case for those ternary strings (among the \(a_n\) strings we first considered) that are palindromes, and determine formulas comparable to those in (1) – (4) above for this special case.
Topological indices of nanotubes are numerical descriptors that are derived from the graph of chemical compounds. Such indices, based on the distances in the graph, are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Szeged index is obtained as a bond additive quantity, where bond contributions are given as the product of the number of atoms closer to each of the two end points of each bond. In this paper, we find an exact expression for the Szeged index of an armchair polyhex nanotube \((TUAC_6{[p,k]}\)).
It is widely recognized that certain graph-theoretic extremal questions play a major role in the study of communication network vulnerability. These extremal problems are special cases of questions concerning the realizability of graph invariants. We define a CS(\(p, q, \lambda, \delta\)) graph as a connected, separable graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). In this notation, if the “CS” is omitted the graph is not necessarily connected and separable. An arbitrary quadruple of integers \((a, b, c, d)\) is called CS(\(p, q, A, 5\)) realizable if there is a CS(\(p, q, \lambda, \delta\)) graph with \(p = a, q = b, \lambda = c\) and \(\delta= d\). Necessary and sufficient conditions for a quadruple to be CS(\(p, q,\lambda, \delta\)) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta), (p, q, \lambda, \Delta), (p, q, \delta, \Delta), (p, q, \lambda, \delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(A\) denotes the maximum degree for all points in a graph and \(\lambda\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa), (p, q, \lambda), (p, q, \delta), (p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.
We use \(k\)-trees to generalize the sequence of Motzkin numbers and show that Baxter’s generalization of Temperley-Lieb operators is a special case of our generalization of Motzkin numbers. We also obtain a recursive summation formula for the terms of \(3\)-Motzkin numbers and investigate some asymptotic properties of the terms of \(k\)-Motzkin numbers.
In this article, defining the matrix extensions of the Fibonacci and Lucas numbers, we start a new approach to derive formulas for some integer numbers which have appeared, often surprisingly, as answers to intricate problems, in conventional and in recreational Mathematics. Our approach provides a new way of looking at integer sequences from the perspective of matrix algebra, showing how several of these integer sequences relate to each other.
For a finite group \(G\) the commutativity degree,
\[d(G)=\frac{|\{(x,y)|x,y \in G, xy=yx\}|}{|G|^2}\]
is defined and studied by several authors and when \(d(G) \geq \frac{1}{2}\) it is proved by P. Lescot in 1995 that \(G\) is abelian , or \(\frac{G}{Z(G)}\) is elementary abelian with \(|G’| = 2\), or \(G\) is isoclinic with \(S_3\) and \(d(G) = 1\). The case when \(d(G) < \frac{1}{2}\) is of interest to study. In this paper we study certain infinite classes of finite groups and give explicit formulas for \(d(G)\). In some cases the groups satisfy \(\frac{1}{4} < d(G) < \frac{1}{2}\). Some of the groups under study are nilpotent of high nilpotency classes.
In this paper, we construct a new infinite family of balanced binary sequences of length \(N = 4p\), \(p \equiv 5 \pmod{8}\) with optimal autocorrelation magnitude \(\{N, 0, \pm 4\}\).
The cocircuits of a splitting matroid \(M_{i,j}\) are described in terms of the cocircuits of the original matroid \(M\).
Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called an \(f\)-factor if \(d_F(x) = f(x)\) for every \(x \in V(F)\). In this paper, we present some sufficient conditions for the existence of \(f\)-factors and connected \((f-2, f)\)-factors in \(K_{1,n}\)-free graphs. The conditions involve the minimum degree, the stability number, and the connectivity of graph \(G\).
We classify the minimal blocking sets of size 15 in \(\mathrm{PG}(2,9)\). We show that the only examples are the projective triangle and the sporadic example arising from the secants to the unique complete 6-arc in \(\mathrm{PG}(2,9)\). This classification was used to solve the open problem of the existence of maximal partial spreads of size 76 in \(\mathrm{PG}(3,9)\). No such maximal partial spreads exist \([13]\). In \([14]\), also the non-existence of maximal partial spreads of size 75 in \(\mathrm{PG}(3,9)\) has been proven. So, the result presented here contributes to the proof that the largest maximal partial spreads in \(\mathrm{PG}(3,q=9)\) have size \(q^2-q+2=74\).
Our work in this paper is concerned with a new kind of fuzzy ideal of a \(K\)-algebra called an \((\in, \in \vee_q)\)-fuzzy ideal. We investigate some interesting properties of \((\in, \in \vee_q)\)-fuzzy ideals of \(K\)-algebras. We study fuzzy ideals with thresholds which is a generalization of both fuzzy ideals and \((\in, \in \vee_q)\)-fuzzy ideals. We also present characterization theorems of implication-based fuzzy ideals.
Let \(G\) be a digraph. For two vertices \(u\) and \(v\) in \(G\), the distance \(d(u,v)\) from \(u\) to \(v\) in \(G\) is the length of the shortest directed path from \(u\) to \(v\). The eccentricity \(e(v)\) of \(v\) is the maximum distance of \(v\) to any other vertex of \(G\). A vertex \(u\) is an eccentric vertex of \(v\) if the distance from \(v\) to \(u\) is equal to the eccentricity of \(v\). The eccentric digraph \(ED(G)\) of \(G\) is the digraph that has the same vertex set as \(G\) and the arc set defined by: there is an arc from \(u\) to \(v\) if and only if \(v\) is an eccentric vertex of \(u\). In this paper, we determine the eccentric digraphs of digraphs for various families of digraphs and we get some new results on the eccentric digraphs of the digraphs.
We present \(3\) open challenges in the field of Costas arrays. They are: a) the determination of the number of dots on the main diagonal of a Welch array, and especially the maximal such number for a Welch array of a given order; b) the conjecture that the fraction of Welch arrays without dots on the main diagonal behaves asymptotically as the fraction of permutations without fixed points and hence approaches \(1/e\) and c) the determination of the parity populations of Golomb arrays generated in fields of characteristic \(2\).
Let \(G\) be the graph obtained from \(K_{3,3}\) by deleting an edge. We find a list assignment with \(|L(v)| = 2\) for each vertex \(v\) of \(G\), such that \(G\) is uniquely \(L\)-colorable, and show that for any list assignment \(L’\) of \(G\), if \(|Z'(v)| \geq 2\) for all \(v \in V(G)\) and there exists a vertex \(v_0\) with \(|L'(v_0)| > 2\), then \(G\) is not uniquely \(L’\)-colorable. However, \(G\) is not \(2\)-choosable. This disproves a conjecture of Akbari, Mirrokni, and Sadjad (Problem \(404\) in Discrete Math. \(266(2003) 441-451)\).
A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. In this note, we show that the vertex set of every graph with minimum degree at least two and with no component that is a \(5\)-cycle can be partitioned into a dominating set and a total dominating set.
Let \(G\) be an undirected graph, \(A\) be an (additive) Abelian group and \(A^* = A – \{0\}\). A graph \(G\) is \(A\)-connected if \(G\) has an orientation such that for every function \(b: V(G) \longmapsto A\) satisfying \(\sum_{v\in V(G)} b(v) = 0\), there is a function \(f: E(G) \longmapsto A^*\) such that at each vertex \(v\in V(G)\) the net flow out of \(v\) equals \(b(v)\). We investigate the group connectivity number \(\Lambda_g(G) = \min\{n; G \text{ is } A\text{-connected for every Abelian group with } |A| \geq n\}\) for complete bipartite graphs, chordal graphs, and biwheels.
Various enumeration problems for classes of simply generated families of trees have been the object of investigation in the past. We mention the enumeration of independent subsets, connected subsets or matchings for instance. The aim of this paper is to show how combinatorial problems of this type can also be solved for rooted trees and trees, which enables us to take better account of isomorphisms. As an example, we will determine the average number of independent vertex subsets of trees and binary rooted trees (every node has outdegree \(\leq 2\)).
In this paper, first we introduce the concept of a \({connected}\) graph homomorphism as a homomorphism for which the inverse image of any edge is either empty or a connected graph, and then we concentrate on chromatically connected (resp. chromatically disconnected) graphs such as \(G\) for which any \(\chi(G)\)-colouring is a connected (resp. disconnected) homomorphism to \(K_{\chi(G)}\).
In this regard, we consider the relationships of the new concept to some other notions as uniquely-colourability. Also, we specify some classes of chromatically disconnected graphs such as Kneser graphs \(KG(m,n)\) for which \(m\) is sufficiently larger than \(n\), and the line graphs of non-complete class II graphs.
Moreover, we prove that the existence problem for connected homomorphisms to any fixed complete graph is an NP-complete problem.
We show that every \(2\)-connected cubic graph of order \(n > 8\) admits a \(P_3\)-packing of at least \(\frac{9n}{11}n\) vertices. The proof is constructive, implying an \(O(M(n))\) time algorithm for constructing such a packing, where \(M(n)\) is the time complexity of the perfect matching problem for \(2\)-connected cubic graphs.
The locally twisted cube \(LTQ_n\) is a newly introduced interconnection network for parallel computing. As a variant of the hypercube \(Q_n\), \(LTQ_n\) has better properties than \(Q_n\) with the same number of links and processors. Yang, Megson and Evans Evans [Locally twisted cubes are \(4\)-pancyclic, Applied Mathematics Letters, \(17 (2004), 919-925]\) showed that \(LTQ_n\) contains a cycle of every length from \(4\) to \(2^n\). In this note, we improve this result by showing that every edge of \(LTQ_n\) lies on a cycle of every length from \(4\) to \(2^n\) inclusive.
Necessary and sufficient conditions are given for the existence of a \((K_3 + e, \lambda)\)-group divisible design of type \(g^tu^1\).
A \(\lambda\)-design on \(v\) points is a set of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(4\)-designs can be obtained from symmetric designs by a complementation procedure. In this paper, we establish feasibility criteria for the existence of \(\lambda\)-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. We use these criteria and a computer to prove that the \(\lambda\)-design conjecture is true for all \(\lambda\)-designs with two block sizes with \(v \leq 90\) and \(\lambda \neq 45\).
In this paper, we consider the relationships between the sums of the Fibonacci and Lucas numbers and \(1\)-factors of bipartite graphs.
We define extended orthogonal sets of \(d\)-cubes and show that they are equivalent to a class of orthogonal arrays, to geometric nets and a class of codes. As a corollary, an upper bound for the maximal number of \(d\)-cubes in an orthogonal set is obtained.
For two given graphs \(G_1\) and \(G_2\), the \({Ramsey\; number}\) \(R(G_1, G_2)\) is the smallest integer \(n\) such that for any graph \(G\) of order \(n\), either \(G\) contains \(G_1\) or the complement of \(G\) contains \(G_2\). Let \(P_n\) denote a path of order \(n\) and \(W_{m}\) a wheel of order \(m+1\). Chen et al. determined all values of \(R(P_n, W_{m})\) for \(n \geq m-1\). In this paper, we establish the best possible upper bound and determine some exact values for \(R(P_n, W_{m})\) with \(n \leq m-2\).
A container \(C(x,y)\) is a set of vertex-disjoint paths between vertices \(z\) and \(y\) in a graph \(G\). The width \(w(C(x,y))\) and length \(L(C(x,y))\) are defined to be \(|C(x,y)|\) and the length of the longest path in \(C(x,y)\) respectively. The \(w\)-wide distance \(d_w(x,y)\) between \(x\) and \(y\) is the minimum of \(L(C(x,y))\) for all containers \(C(x,y)\) with width \(w\). The \(w\)-wide diameter \(d_w(G)\) of \(G\) is the maximum of \(d_w(x,y)\) among all pairs of vertices \(x,y\) in \(G\), \(x \neq y\). In this paper, we investigate some problems on the relations between \(d_w(G)\) and diameter \(d(G)\) which were raised by D.F. Hsu \([1]\). Some results about graph equation of \(d_w(G)\) are proved.
In this paper, we use a genetic algorithm and direct a hill-climbing algorithm in choosing differences to generate solutions for difference triangle sets. The combined use of the two algorithms optimized the hill-climbing method and produced new improved upper bounds for difference triangle sets.
The covering problem in the \( n \)-dimensional \( q \)-ary Hamming space consists of the determination of the minimal cardinality \( K_q(n, R) \) of an \( R \)-covering code. It is known that the sphere covering bound can be improved by considering decompositions of the underlying space, leading to integer programming problems. We describe the method in an elementary way and derive about 50 new computational and theoretical records for lower bounds on \( K_q(n, R) \).
For any graph \( G = (V, E) \), \( D \subseteq V \) is a global dominating set if \( D \) dominates both \( G \) and its complement \( \overline{G} \). The global domination number \( \gamma_g(G) \) of a graph \( G \) is the fewest number of vertices required of a global dominating set. In general,\(
\max\{\gamma(G), \gamma(\overline{G})\} \leq \gamma_g(G) \leq \gamma(G) + \gamma(\overline{G}),\) where \( \gamma(G) \) and \( \gamma(\overline{G}) \) are the respective domination numbers of \( G \) and \( \overline{G} \). We show that when \( G \) is a planar graph, \(\gamma_g(G) \leq \max\{\gamma(G) + 1, 4\}.\)
Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( D \) as a minimum feedback arc set. The reversing number of a digraph is defined to be \(r(D) = |V(T)| – |V(D)|.\)
We use integer programming methods to obtain new results for the reversing number where \( D \) is a power of a directed Hamiltonian path. As a result, we establish that known reversing numbers for certain classes of tournaments actually suffice for a larger class of digraphs.
A directed covering design, \( DC(v, k, \lambda) \), is a \( (v, k, 2\lambda) \) covering design in which the blocks are regarded as ordered \( k \)-tuples and in which each ordered pair of elements occurs in at least \( \lambda \) blocks. Let \( DE(v, k, \lambda) \) denote the minimum number of blocks in a \( DC(v, k, \lambda) \). In this paper, the values of the function \( DE(v, k, \lambda) \) are determined for all odd integers \( v \geq 5 \) and \( \lambda \) odd, with the exception of \( (v, \lambda) = (53, 1), (63, 1), (73, 1), (83, 1) \). Further, we provide an example of a covering design that cannot be directed.
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). For a labeling \( f: V(G) \to A = \{0, 1\} \), define a partial edge labeling \( f^*: E(G) \to A \) such that, for each edge \( xy \in E(G) \),\(f^*(xy) = f(x) \quad \text{if and only if} \quad f(x) = f(y).\) For \( i \in A \), let \(\text{v}_f(i) = |\{ v \in V(G) : f(v) = i \}|\) and \(\text{e}_{f^*}(i) = |\{ e \in E(G) : f^*(e) = i \}|.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(
|\text{v}_f(0) – \text{v}_f(1)| \leq 1.\) If a friendly labeling \( f \) induces a partial labeling \( f^* \) such that \(|\text{e}_{f^*}(0) – \text{e}_{f^*}(1)| \leq 1,\)then \( G \) is said to be balanced. In this paper, a necessary and sufficient condition for balanced graphs is established. Using this result, the balancedness of several families of graphs is also proven.
Expanding upon a comment by P. A. Leonard [9], we exhibit \(\mathbb{Z}\)-cyclic patterned-starter based whist tournaments for \(q^2\) players, where \(g = 4k + 3\) is prime; the cases \(3 < q < 200\) are included herein, with data for \(200 < q < 5,000\) available electronically.
Let \( G \) be a \( (p, q) \)-graph and \( k \geq 0 \). A graph \( G \) is said to be k-edge-graceful if the edges can be labeled by \( k, k+1, \dots, k+q-1 \) so that the vertex sums are distinct, modulo \( p \). We denote the set of all \( k \) such that \( G \) is \( k \)-edge graceful by \( \text{egS}(G) \). The set is called the \textbf{edge-graceful spectrum} of \( G \). In this paper, we are concerned with the problem of exhibiting sets of natural numbers which are the edge-graceful spectra of the cylinder \( C_{n} \times P_{m} \), for certain values of \( n \) and \( m \).
The judgment aggregation problem is an extension of the group decision-making problem, wherein each voter votes on a set of propositions which may be logically interrelated (such as \( p \), \( p \to q \), and \( q \)). The simple majority rule can yield an inconsistent set of results, so more complicated rules must be developed. Here, the problem is cast in terms of matroids, and the Greedy Algorithm is modified to obtain a “best” result. An NP-completeness result is also presented for this particular formulation of the problem.
Let \( G \) be a connected simple \( (p, q) \)-graph and \( k \) a non-negative integer. The graph \( G \) is said to be \( k \)-edge-graceful if the edges can be labeled with \( k, k+1, \dots, k+q-1 \) so that the vertex sums are distinct modulo \( p \). The set of all \( k \) where \( G \) is \( k \)-edge-graceful is called the edge-graceful spectrum of \( G \). In 2004, Lee, Cheng, and Wang analyzed the edge-graceful spectra of certain connected bicyclic graphs, leaving some cases as open problems. Here, we determine the edge-graceful spectra of all connected bicyclic graphs without pendant.
Let \( G \) be a connected graph with vertex set \( V(G) \) and edge set \( E(G) \). A (defensive) alliance in \( G \) is a subset \( S \) of \( V(G) \) such that for every vertex \( v \in S \),\(|N[v] \cap S| \geq |N(v) \cap (V(G) – S)|.\) The alliance partition number, \( \psi_a(G) \), was defined (and further studied in [11]) to be the maximum number of sets in a partition of \( V(G) \) such that each set is a (defensive) alliance. Similarly, \( \psi_g(G) \) is the maximum number of sets in a partition of \( V(G) \) such that each set is a global alliance, i.e., each set is an alliance and a dominating set. In this paper, we give bounds for the global alliance partition number in terms of the minimum degree, which gives exactly two values for \( \psi_g(G) \) in trees. We concentrate on conditions that classify trees to have \( \psi_g(G) = i \) (\( i = 1, 2 \)), presenting a characterization for binary trees.
E. Stickel proposed a variation of the Diffie-Hellman key exchange scheme based on non-abelian groups, claiming that the underlying problem is more secure than the traditional discrete logarithm problem in cyclic groups. We show that the proposed scheme does not provide a higher level of security in comparison to the traditional Diffie-Hellman scheme.
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \(f^*(xy) = f(x), \text{ if and only if } f(x) = f(y),\) for each edge \( xy \in E(G) \). For \( i \in A \), let \(v_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}\) and \(e_f^*(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(|v_f(0) – v_f(1)| \leq 1.\) If \(|e_f(0) – e_f(1)| \leq 1,\)then \( G \) is said to be \textbf{\emph{balanced}}. The \textbf{\emph{balance index set}} of the graph \( G \), \( BI(G) \), is defined as \(BI(G) = \{ |e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly} \}.\)Results parallel to the concept of friendly index sets are pr
An orthogonal double cover (ODC) of the complete graph \( K_n \) by a graph \( G \) is a collection \( \mathcal{G} = \{G_i \mid i=1,2,\dots,n\} \) of spanning subgraphs of \( K_n \), all isomorphic to \( G \), with the property that every edge of \( K_n \) belongs to exactly two members of \( \mathcal{G} \) and any two distinct members of \( \mathcal{G} \) share exactly one edge.
A lobster of diameter five is a tree arising from a double star by attaching any number of pendant vertices to each of its vertices of degree one. We show that for any double star \( R(p, q) \) there exists an ODC of \( K_n \) by all lobsters of diameter five (with finitely many possible exceptions) arising from \( R(p, q) \).
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces an edge partial labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in A \), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|.\)The balance index set of \( G \), denoted \( BI(G) \), is defined as \(\{|e_f(0) – e_f(1)|: |v_f(0) – v_f(1)| \leq 1\}.\)In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.
For any \( h \in \mathbb{Z} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by
\[
l^+(v) = \sum_{uv \in E(G)} l(uv)\]
is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will determine the integer-magic spectra of trees of diameter five.
A graceful labeling of a directed graph \( D \) with \( e \) edges is a one-to-one map \( \theta: V(D) \to \{0, 1, \dots, e\} \) such that \( \theta(y) – \theta(x) \mod (e + 1) \) is distinct for each \( (x, y) \in E(D) \). This paper summarizes previously known results on graceful directed graphs and presents some new results on directed paths, stars, wheels, and umbrellas.
For an integer \( l > 1 \), the \( l \)-edge-connectivity of a graph \( G \) with \( |V(G)| \geq l \), denoted by \( \lambda_l(G) \), is the smallest number of edges whose removal results in a graph with \( l \) components. In this paper, we study lower bounds of \( \lambda_l(G) \) and optimal graphs that reach the lower bounds. Former results by Boesch and Chen are extended.
We also present in this paper an optimal model of interconnection network \( G \) with a given \( \lambda_l(G) \) such that \( \lambda_2(G) \) is maximized while \( |E(G)| \) is minimized.
Given an abelian group \( A \), a graph \( G = (V, E) \) is said to have a distance two magic labeling in \( A \) if there exists a labeling \( l: E(G) \to A – \{0\} \) such that the induced vertex labeling \( l^*: V(G) \to A \) defined by
\[l^*(v) = \sum_{c \in E(v)} l(e)\]
is a constant map, where \( E(v) = \{e \in E(G) : d(v,e) < 2\} \). The set of all \( h \in \mathbb{Z}_+ \), for which \( G \) has a distance two magic labeling in \( \mathbb{Z}_h \), is called the distance two magic spectrum of \( G \) and is denoted by \( \Delta M(G) \). In this paper, the distance two magic spectra of certain classes of graphs will be determined.
In this paper, we derive some necessary existence conditions for a bi-level balanced array (B-array) with strength \( t = 5 \). We then describe how these existence conditions can be used to obtain an upper bound on the number of constraints of these arrays, and give some illustrative examples to this effect.
Let \( G = (V, E) \) be a graph with a vertex labeling \( f: V \to \mathbb{Z}_2 \) that induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \). For each \( i \in \mathbb{Z}_2 \), let \(
v_f(i) = \text{card}\{v \in V: f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E: f^*(e) = i\}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\) The friendly index set of \( G \) is defined as \(\{\lvert e_f(1) – e_f(0) \rvert : \text{the vertex labeling } f \text{ is friendly}\}.\)
In this paper, we determine the friendly index sets of generalized books.
Given 2 triangles in a plane over a field \( F \) which are in perspective from a vertex \( V \), the resulting Desargues line or axis \( l \) may or may not be on \( V \). To avoid degenerate cases, we assume that the union of the vertices of the 2 triangles is a set of six points with no three collinear. Our work then provides a detailed analysis of situations when \( V \) is on \( l \) for any \( F \), finite or infinite.
We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given. For example, the nonexistence of the graphs with \( n = 11 \) and degree sequence \( (5, 5, 5, \ldots, 4) \) and \( n = 13 \) and degree sequence \( (6, 5, 5, \ldots, 5) \) are shown.
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). Let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \(f^*(xy) = f(x) \quad \text{if and only if } f(x) = f(y),\) for each edge \( xy \in E(G) \). For \( i \in A \), let \(
v_f(i) = \text{card}\{v \in V(G) : f(v) = i\}\) and \(e_{f^*}(i) = \text{card}\{e \in E(G) : f^*(e) = i\}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\)If \(\lvert e_{f^*}(0) – e_{f^*}(1) \rvert \leq 1,\) then \( G \) is said to be \(\textbf{balanced}\). The balancedness of the Cartesian product and composition of graphs is studied in [19]. We provide some new families of balanced graphs using other constructions.
Greedy defining sets have been studied for the first time by the author for graphs. In this paper, we consider greedy defining sets for Latin squares and study the structure of these sets in Latin squares. We give a general bound for greedy defining numbers and linear bounds for greedy defining numbers of some infinite families of Latin squares. Greedy defining sets of circulant Latin squares are also discussed in the paper.
Let \(C_n^{(t)}\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 7, 9, 4k\). In this paper, the conjecture is shown to be true for \(n = 11\).
Let \(P(G; \lambda)\) denote the chromatic polynomial of a graph \(G\), expressed in the variable \(\lambda\). Then \(G\) is said to be chromatically unique if \(G\) is isomorphic with \(H\) for any graph \(H\) such that \(P(H; \lambda) = P(G; \lambda)\). The graph consisting of \(s\) edge-disjoint paths joining two vertices is called an \(s\)-bridge graph. In this paper, we provide a new family of chromatically unique \(5\)-bridge graphs.
Recently in \([5]\), the author considered certain reciprocal sums of general second order recurrence \(\{W_n\}\). In this paper, we generalize the results of Xi and we give some new results for the reciprocal sums of \(k\)-th power of general second order recurrence \(\{W_{kn}\}\) for arbitrary positive integer \(k\).
In this article, we study the generalized Bernoulli and Euler polynomials, and obtain relationships between them, based upon the technique of matrix representation.
Let \(K_v\) be the complete graph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-decomposition of \(K_v\), denoted by \(G\)-GD\((v)\), is a pair \((X, \mathcal{B})\) where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, nine graphs \(G_i\) with six vertices and nine edges are discussed, and the existence of \(G_i\)-decompositions are completely solved, \(1 \leq i \leq 9\).
A graph \(G\) is called \({uniquely\; k-list \;colorable}\), or \(UkLC\) for short, if it admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring. A graph \(G\) is said to have the property \(M(k)\) (M for Marshal Hall) if and only if it is not \(UkLC\). In \(1999\), M. Ghebleh and E.S. Mahmoodian characterized the \(U3LC\) graphs for complete multipartite graphs except for nine graphs. At the same time, for the nine exempted graphs, they give an open problem: verify the property \(M(3)\) for the graphs \(K_{2,2,\ldots,2}\) for \(r = 4,5,\ldots,8\), \(K_{2,3,4}\), \(K_{1*4,4}\), \(K_{1*4,4}\), and \(K_{1*5,4}\). Until now, except for \(K_{1*5,4}\), the other eight graphs have been showed to have the property \(M(3)\) by W. He et al. In this paper, we show that graph \(K_{1*5,4}\) has the property \(M(3)\), and as consequences, \(K_{1*4,4}\), \(K_{2,2,4}\) have the property \(M(3)\). Therefore the \(U3LC\) complete multipartite graphs are completely characterized.
Given a graph \(G\), we say \(S \subseteq V(G)\) is \({resolving}\) if for each pair of distinct \(u, v \in V(G)\) there is a vertex \(x \in S\) where \(d(u, x) \neq d(v, x)\). The metric dimension of \(G\) is the minimum cardinality of all resolving sets. For \(w \in V(G)\), the distance from \(w\) to \(S\), denoted \(d(w, S)\), is the minimum distance between \(w\) and the vertices of \(S\). Given \(\mathcal{P} = \{P_1, P_2, \ldots, P_k\}\) an ordered partition of \(V(G)\), we say \(P\) is resolving if for each pair of distinct \(u, v \in V(G)\) there is a part \(P_i\) where \(d(u, P_i) \neq d(v, P_i)\). The partition dimension is the minimum order of all resolving partitions. In this paper, we consider relationships between metric dimension, partition dimension, diameter, and other graph parameters. We construct “universal examples” of graphs with given partition dimension, and we use these to provide bounds on various graph parameters based on metric and partition dimensions. We form a construction showing that for all integers \(a\) and \(b\) with \(3 \leq a \leq \beta + 1\), there exists a graph \(G\) with partition dimension \(\alpha\) and metric dimension \(\beta\), answering a question of Chartrand, Salehi, and Zhang \([3]\).
The total chromatic number \(\chi_\tau(G)\) is the least number of colours needed to colour the vertices and edges of a graph \(G\) such that no incident or adjacent elements (vertices or edges) receive the same colour. This work determines the total chromatic number of grids, particular cases of partial grids, near-ladders, and of \(k\)-dimensional cubes.
The \({star arboricity}\) \(sa(G)\) of a graph \(G\) is the minimum number of star forests which are needed to decompose all edges of \(G\). For integers \(k\) and \(n\), \(1 \leq k \leq n\), the \({crown}\) \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : i = 0, 1, \ldots, n-1, j \equiv i+1, i+2, \ldots, i+k \pmod{n}\}\). In \([2]\), Lin et al. conjectured that for every \(k\) and \(n\), \(3 \leq k \leq n-1\), the star arboricity of the crown \(C_{n,k}\) is \(\lceil k/2 \rceil + 1\) if \(k\) is odd and \(\lceil k/2 \rceil + 2\) otherwise. In this note, we show that the above conjecture is not true for the case \(n = 9t\) (\(t\) is a positive integer) and \(k = 4\) by showing that \(sa(C_{9t,4}) = 3\).
Let \(\mathcal{P}(n,k)\) denote the number of graphs on \(n+k\) vertices that contain \(P_n\), a path on \(n\) vertices, as an induced subgraph. In this note, we will find upper and lower bounds for \(\mathcal{P}(n,k)\). Using these bounds, we show that for \(k\) fixed, \(\mathcal{P}(n,k)\) behaves roughly like an exponential function of \(n\) as \(n\) gets large.
A \({dominating \;broadcast}\) of a graph \(G\) of diameter \(d\) is a function \(f: V(G) \to \{0, 1, 2, \ldots, d\}\) such that for all \(v \in V(G)\) there exists \(u \in V(G)\) with \(d(u, v) \leq f(u)\). We investigate dominating broadcasts for caterpillars.
In this paper, we obtain a fundamental result on the dynamical behavior of symmetric weighted mappings for two-dimensional real sequence spaces \({R}_s\).
In \(2006\), Mojdeh and Jafari Rad [On the total domination critical graphs, Electronic Notes in Discrete Mathematics, 24 (2006), 89-92] gave an open problem: Does there exist a \(3\)-\(\gamma_t\)-critical graph \(G\) of order \(\Delta(G) + 3\) with \(\Delta(G)\) odd and \(\delta(G) \geq 2\)? In this paper, we positively answer that for each odd integer \(n \geq 9\), there exists a \(3\)-\(\gamma_t\)-critical graph \(G_n\) of order \(n+3\) with \(\delta(G) \geq 2\). On the contrary, we also prove that for \(\Delta(G) = 3, 5, 7\), there is no \(3\)-\(\gamma_t\)-critical graph of order \(\Delta(G) + 3\) with \(\delta(G) \geq 2\).
Let \(\{w_n\}\) be a second-order recurrence sequence. According to the definition and characteristics of the recurrent sequence, we proved a recursion formula for certain reciprocal sums whose denominators are products of consecutive elements of \(\{w_n\}\).
Let \(G\) be a graph in which each vertex has been colored using one of \(k\) colors, say \(c_1, c_2, \ldots, c_k\). If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices colored \(c_i\), \(i = 1, 2, \ldots, k\), and \(|n_i – n_j| \leq 1\) for any \(i, j \in \{1, 2, \ldots, k\}\), then \(C\) is equitably \(k\)-colored. An \(m\)-cycle decomposition \(\mathcal{C}\) of a graph \(G\) is equitably \(k\)-colorable if the vertices of \(G\) can be colored so that every \(m\)-cycle in \(\mathcal{C}\) is equitably \(k\)-colored. For \(m = 4, 5\), and \(6\), we completely settle the existence problem for equitably \(2\)-colorable \(m\)-cycle decompositions of complete graphs with the edges of a \(1\)-factor added.
Suppose a network facility location problem is modelled by means of an undirected, simple graph \(G = (\mathcal{V, E})\) with \(\mathcal = \{v_1, \ldots, v_n\}\). Let \(\mathbf{r} = (r_1, \ldots, r_n)\) and \(\mathbf{s} = (s_1, \ldots, s_n)\) be vectors of nonnegative integers and consider the combinatorial optimization problem of locating the minimum number, \(\gamma(\mathbf{r}, \mathbf{s}, G)\) (say), of commodities on the vertices of \(G\) such that at least \(s_j\) commodities are located in the vicinity of (i.e. in the closed neighbourhood of) vertex \(v_j\), with no more than \(r_j\) commodities placed at vertex \(v_j\) itself, for all \(j = 1, \ldots, n\). In this paper we establish lower and upper bounds on the parameter \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a general graph \(G\). We also determine this parameter exactly for certain classes of graphs, such as paths, cycles, complete graphs, complete bipartite graphs and establish good upper bounds on \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a class of grid graphs in the special case where \(r_j = r\) and \(s_j = s\) for all \(j = 1, \ldots, n\).
Let \(A\) be an arbitrary circulant stochastic matrix, and let \(\underline{x}_0\) be a vector. An “asymptotic” canonical form is derived for \(A^k\) (as \(k \to \infty\)) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with transition matrix \(A\), and by analyzing how \(A\) acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of \(A^k\) is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence \(\underline{x}_k = A^k \underline{x}_0\), is accomplished by utilizing congruential matrix identities and \((0,1)\)-matrices to determine the minimum \(2\)-adic order of the coordinates of \(\underline{x}_k\), through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeatedly applied simultaneously to each term of a finite sequence when read cyclically.
A graph \(G\) is \(s\)-Hamiltonian if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) has a Hamiltonian cycle, and \(s\)-Hamiltonian connected if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) is Hamiltonian-connected. Let \(k > 0, s \geq 0\) be two integers. The following are proved in this paper:(1) Let \(k \geq s+2\) and \(s \leq n-3\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s)/2\) for every independent set \(I\) of order \(k-s\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s-1\), then \(G\) is \(s\)-Hamiltonian.(2) Let \(k \geq s+3\) and \(s \leq n-2\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s+1)/2\) for every independent set \(I\) of order \(k-s-1\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s\), then \(G\) is \(s\)-Hamiltonian connected.These results extend several former results by Dirac, Ore, Fan, and Chen.
We show that every generalized quadrangle of order \((4,6)\) with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle \(\text{AS}(5)\). It then easily follows that every generalized quadrangle of order \(5\) with an axis of symmetry is isomorphic to the classical generalized quadrangle \(\text{Q}(4, 5)\).
A \(C_5C_7\) net is a trivalent decoration made by alternating pentagons \(C_5\) and heptagons \(C_7\). It can cover either a cylinder or a torus. In this paper, we compute the Szeged index of \(HC_5C_7[ r, p ]\) nanotube.
We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types \(C, D\), and \(L\). Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.
The number of essentially different square polyominoes of order \(n\) and minimum perimeter \(p(n)\) is enumerated.
Let \(G = (V, E)\) be a graph. Then \(S \subseteq V\) is an excess-\(t\) global powerful alliance if \(|N[v] \cap S| \geq |N[v] \cap (V – S)| + t\) for every \(v \in V\). If \(t = 0\), this definition reduces to that of a \({global \;powerful \;alliance}\). Here we determine bounds on the cardinalities of such sets \(S\).
A total perfect code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is adjacent to exactly one vertex in the subset. We prove that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code.
We calculate the norm of weighted composition operators \(uC_\psi\) from the Bloch space to the weighted space \(H^\infty_\mu({B})\) on the unit ball \({B}\).
Let \(P\) be a polygon whose vertices have been colored (labeled) cyclically with the numbers \(1, 2, \ldots, c\). Motivated by conjectures of Propp, we are led to consider partitions of \(P\) into \(k\)-gons which are proper in the sense that each \(k\)-gon contains all \(c\) colors on its vertices. Counting the number of proper partitions involves a generalization of the \(k\)-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.
Let \(n, k\) be integers and \(k < n\). Denote by \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) the set of graphs of order \(n\) with \(k\) independent vertices and the set of graphs of order \(n\) with \(k\) independent edges, respectively. The bounds of the spectral radius of graphs in \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) are obtained.
Let \(n \in \mathbb{N}\) and let \(A \subseteq \mathbb{Z}_n\) be such that \(A\) does not contain \(0\) and is non-empty. We define \({E}_A(n)\) to be the least \(t \in \mathbb{N}\) such that for all sequences \((x_1, \ldots, x_t) \in \mathbb{Z}^t\), there exist indices \(j_1, \ldots, j_n \in \mathbb{N}\), \(1 \leq j_1 < \cdots < j_n \leq t\), and \((\theta_1, \ldots, \theta_n) \in A^n\) with \(\sum_{i=1}^n \theta_i x_{j_i} \equiv 0 \pmod{n}\). Similarly, for any such set \(A\), we define the \({Davenport Constant}\) of \(\mathbb{Z}_n\) with weight \(A\) denoted by \(D_A(n)\) to be the least natural number \(k\) such that for any sequence \((x_1, \ldots, x_k) \in \mathbb{Z}^k\), there exist a non-empty subsequence \((x_{j}, \ldots, x_{j_i})\) and \((a_1, \ldots, a_l) \in A^t\) such that \(\sum_{i=1}^n a_i x_{j_i} \equiv 0 \pmod{n}\). Das Adhikari and Rath conjectured that for any set \(A \subseteq \mathbb{Z}_n \setminus \{0\}\), the equality \({E}_A(n) = D_A(n) + n – 1\) holds. In this note, we determine some Davenport constants with weights and also prove that the conjecture holds in some special cases.
In this paper, we introduce an extension of the hyperbolic Fibonacci and Lucas functions which were studied by Stakhov and Rozin. Namely, we define hyperbolic functions by second-order recurrence sequences and study their hyperbolic and recurrence properties. We give the corollaries for Fibonacci, Lucas, Pell, and Pell-Lucas numbers. We finalize with the introduction of some surfaces (the Metallic Shofars) that relate to the hyperbolic functions with the second-order recurrence sequences.
The graph’s irregularity is the sum of the absolute values of the differences of degrees of pairs of adjacent vertices in the graph. We provide various upper bounds for the irregularity of a graph, especially for \(K_{r+1}\)-free graphs, where \(K_{r+1}\) is a complete graph on \(r+1\) vertices, and trees and unicyclic graphs of given number of pendant vertices.
Let \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) be the \(n\)-dimensional vector (resp. affine) space over the finite field \(\mathbb{F}_q\). For \(1 \leq i \leq i+s \leq n-1\) (resp. \(0 \leq i \leq i+s \leq n-1\)), let \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) denote the set of all subspaces (resp. flats) in \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) with dimensions between \(i\) and \(i+s\) including \(\mathbb{F}_q^(n)\) and \(\{0\}\) (resp. \(\emptyset\)). By ordering \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) by ordinary inclusion or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity.
In this paper, we give some relations involving the usual Fibonacci and generalized order-\(k\) Pell numbers. These relations show that the generalized order-\(k\) Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We find families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, \(F_{2i-1}\), and their sums. Also, extending these matrix representations, we find families of super-diagonal matrices such that the permanents of these matrices are the generalized order-\(k\) Pell numbers and their sums.
Let \(G\) be a finite group and \(S\) be a subset (possibly containing the identity element) of \(G\). We define the Bi-Cayley graph \(X = BC(G, S)\) to be the bipartite graph with vertices \(G \times \{0, 1\}\) and edges \(\{(g, 0), (sg, 1) : g \in G, s \in S\}\). In this paper, we show that if \(X = BC(G, S)\) is connected, then \(\kappa(X) = \delta(X)\).
Some new characterizations for harmonic Bergman space on the unit ball \({B}\) in \(\mathbb{R}^n\) are given in this paper. They can be described as derivative-free characterizations.
The planar Ramsey number \(PR(H_1, H_2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains a copy of \(H_1\) or its complement contains a copy of \(H_2\). It is known that the Ramsey number \(R(K_4 – e, K_k – e)\) for \(k \leq 6\). In this paper, we prove that \(PR(K_4 – e, K_6 – e) = 16\) and show the lower bounds on \(PR(K_4 – e, K_k – e)\).
In this paper, we extend the idea of magic labeling to directed graphs. In particular, a magic labeling of a digraph is the directed analog of a vertex-magic total labeling. Some elementary results are obtained and some infinite families of magic digraph labelings are exhibited.
Let \( P_h \) be a path on \( h \) vertices. A simple graph \( G = (V, E) \) admits a \( P_h \)-covering if every edge in \( E \) belongs to a subgraph of \( G \) that is isomorphic to \( P_h \). \( G \) is called \( P_h \)-magic if there is a total labeling \( f: V \cup E \to \{1, 2, \dots, |V| + |E|\} \) such that for each subgraph \( H’ = (V’, E’) \) of \( G \) that is isomorphic to \( P_h \), \( \sum_{v \in V’} f(v) + \sum_{e \in E’} f(e) \) is constant. When \( f(V) = \{1, 2, \dots, |V|\} \), we say that \( G \) is \( P_h \)-supermagic.
In this paper, we study some \( P_h \)-supermagic trees. We give some sufficient or necessary conditions for a tree to be \( P_h \)-supermagic. We also consider the \( P_h \)-supermagicness of special types of trees, namely shrubs and banana trees.
A \((p, q)\)-graph \( G \) is said to be multiplicative if its vertices can be assigned distinct positive integers so that the values of the edges, obtained as the products of the numbers assigned to their end vertices, are all distinct. Such an assignment is called a multiplicative labeling of \( G \). A multiplicative labeling is said to be \((a, r)\)-geometric if the values of the edges can be arranged as a geometric progression \( a, ar, ar^2, \dots, ar^{q-1} \). In this paper, we prove that some well-known classes of graphs are geometric for certain values of \( a, r \) and also initiate a study on the structure of finite \((a, r)\)-geometric graphs.
Given an integer \( \lambda \geq 2 \), a graph \( G = (V, E) \) and a spanning subgraph \( H \) of \( G \) (the backbone of \( G \)), a \( \lambda \)-backbone coloring of \( (G, H) \) is a proper vertex coloring \( V \to \{1, 2, \dots\} \) of \( G \), in which the colors assigned to adjacent vertices in \( H \) differ by at least \( \lambda \). We study the computational complexity of the problem “Given a graph \( G \) with a backbone \( H \), and an integer \( \ell \), is there a \( \lambda \)-backbone coloring of \( (G, H) \) with at most \( \ell \) colors?” Of course, this general problem is NP-complete. In this paper, we consider this problem for collections of pairwise disjoint complete graphs with order \( n \). We show that the complexity jumps from polynomially solvable to NP-complete between \( \ell = (n – 1)\lambda \) and \( \ell = (n – 1)\lambda + 1 \).
For a simple graph \( G = (V(G), E(G)) \) with the vertex set \( V(G) \) and the edge set \( E(G) \), a labeling \( \lambda: V(G) \cup E(G) \to \{1, 2, \dots, k\} \) is called an edge-irregular total \( k \)-labeling of \( G \) if for any two different edges \( e = e_1e_2 \) and \( f = f_1f_2 \) in \( E(G) \) we have \( wt(e) \neq wt(f) \) where \( wt(e) = \lambda(e_1) + \lambda(e) + \lambda(e_2) \). The total edge-irregular strength, denoted by \( tes(G) \), is the smallest positive integer \( k \) for which \( G \) has an edge-irregular total \( k \)-labeling. In this paper, we determine the total edge-irregular strength of the corona product of paths with some graphs.
For two given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the smallest positive integer \( N \) such that for every graph \( F \) of order \( N \) the following holds: either \( F \) contains \( G \) as a subgraph or the complement of \( F \) contains \( H \) as a subgraph. In this paper, we shall study the Ramsey number \( R(T_n, W_m) \) for a star-like tree \( T_n \) with \( n \) vertices and a wheel \( W_m \) with \( m + 1 \) vertices and \( m \) odd. We show that the Ramsey number \( R(S_n, W_m) = 3n – 2 \) for \( n \geq 2m – 4, m \geq 5 \) and \( m \) odd, where \( S_n \) denotes the star on \( n \) vertices. We conjecture that the Ramsey number is the same for general trees on \( n \) vertices, and support this conjecture by proving it for a number of star-like trees.
A simple graph \( G(V, E) \) is called \( A \)-magic if there is a labeling \( f: E \to A^* \), where \( A \) is an Abelian group and \( A^* = A – \{0\} \), such that the induced vertex labeling \( f^*: V \to A \), defined as \( f^*(v) = \sum_{u \in N(v)} f(uv) = k \), for every \( v \in V \), is a constant in \( A \). In this paper, we show constructions of new classes of \( A \)-magic graphs from known \( A \)-magic graphs using labeling matrices.
For an ordered set \( W = \{w_1, w_2, \dots, w_k\} \) of vertices and a vertex \( v \) in a connected graph \( G \), the representation of \( v \) with respect to \( W \) is the ordered \( k \)-tuple \( r(v|W) = (d(v, w_1), d(v, w_2), \dots, d(v, w_k)) \) where \( d(x,y) \) represents the distance between the vertices \( x \) and \( y \). The set \( W \) is called a resolving set for \( G \) if every two vertices of \( G \) have distinct representations. A resolving set containing a minimum number of vertices is called a basis for \( G \). The dimension of \( G \), denoted by \( \text{dim}(G) \), is the number of vertices in a basis of \( G \). In this paper, we determine the dimensions of some corona graphs \( G \odot K_1 \), \( G \odot \overline{K}_m \) for any graph \( G \) and \( m \geq 2 \), and a graph with pendant edges more general than corona graphs \( G \odot \overline{K}_m \).
Let \( G = (V, E) \) be a simple, finite, and undirected graph. A sum labeling is a one-to-one mapping \( L \) from a set of vertices of \( G \) to a finite set of positive integers \( S \) such that if \( u \) and \( v \) are vertices of \( G \), then \( uv \) is an edge in \( G \) if and only if there is a vertex \( w \) in \( G \) and \( L(w) = L(u) + L(v) \). A graph \( G \) that has a sum labeling is called a sum graph. The minimal isolated vertex that is needed to make \( G \) a sum labeling is called the sum number of \( G \), denoted as \( \sigma(G) \). The sum number of a sum graph \( G \) is always greater than or equal to \( \delta(G) \), the minimum degree of \( G \). An optimum sum graph is a sum graph that has \( \sigma(G) = \delta(G) \). In this paper, we discuss sum numbers of finite unions of some families of optimum sum graphs, such as cycles and friendship graphs.
Let \( G = (V, E) \) be a simple and undirected graph with \( v \) vertices and \( e \) edges. An \( (a, d) \)-\({edge-antimagic\; total\; labeling}\) is a bijection \( f \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( \{1, 2, \dots, v + e\} \) such that the weights of the edges form an arithmetic progression with initial term \( a \) and common difference \( d \). A super \( (a, d) \)-\({edge\; antimagic \;total \;labeling}\) is an edge antimagic total labeling \( f \) such that \( f(V(G)) = \{1, \dots, v\} \). In this paper, we solve some problems on edge antimagic total labeling, such as on paths and unicyclic graphs.
We investigate the critical set of edge-magic labeling on caterpillar graphs and its application on secret sharing schemes. We construct a distribution scheme based on supervisory secret sharing schemes, which use the notion of critical sets to distribute the shares and reconstruct the key.
For given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the least natural number \( n \) such that for every graph \( F \) of order \( n \) the following condition holds: either \( F \) contains \( G \) or the complement of \( F \) contains \( H \). In this paper, we improve the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number \( R(\cup G, H) \), where \( G \) is a path and \( H \) is a Jahangir graph.
A total vertex irregular labeling of a graph G with v vertices and e edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct.The total vertex irregularity strength of \(G\), denoted by \(tvs(G)\), is the minimum value of the largest label over all such irregular assignments.In this paper, we consider the total vertex irregular labelings of wheels W_n, fans \(F_n\), suns \(S_n\) and friendship graphs \(f_n\).We show that \(tvs(W_n) = \lceil \frac{n+3}{4} \rceil \text{ for } n \geq 3\),\(tvs(F_n) = \lceil \frac{n+2}{4} \rceil \text{ for } n \geq 3\),\(tvs(S_n) = \lceil\frac{n+1}{2} \rceil \text{ for } n \geq 3\),\(tvs(f_n) = \lceil \frac{2n+2}{3} \rceil \text{ for all } n\).
For any given graphs \( G \) and \( H \), we write \( F \rightarrow (G, H) \) to mean that any red-blue coloring of the edges of \( F \) contains a red copy of \( G \) or a blue copy of \( H \). A graph \( F \) is \((G, H)\)-minimal (Ramsey-minimal) if \( F \rightarrow (G, H) \) but \( F^* \not\rightarrow (G, H) \) for any proper subgraph \( F^* \subset F \). The class of all \((G, H)\)-minimal graphs is denoted by \( \mathcal{R}(G, H) \). In this paper, we will determine the graphs in \( \mathcal{R}(K_{1,2}, C_4) \).
Let \( G \) be a connected graph. For a vertex \( v \in V(G) \) and an ordered \( k \)-partition \( \Pi = (S_1, S_2, \dots, S_k) \) of \( V(G) \), the representation of \( v \) with respect to \( \Pi \) is the \( k \)-vector \( r(v|\Pi) = (d(v, S_1), d(v, S_2), \dots, d(v, S_k) ) \) where \( d(v, S_i) = \min_{w \in S_i} d(x, w) \) (\( 1 \leq i \leq k \)). The \( k \)-partition \( \Pi \) is said to be resolving if the \( k \)-vectors \( r(v|\Pi) \), \( v \in V(G) \), are distinct. The minimum \( k \) for which there is a resolving \( k \)-partition of \( V(G) \) is called the partition dimension of \( G \), denoted by \( pd(G) \). A resolving \( k \)-partition \( \Pi = \{ S_1, S_2, \dots, S_k \} \) of \( V(G) \) is said to be connected if each subgraph \( \langle S_i \rangle \) induced by \( S_i \) (\( 1 \leq i \leq k \)) is connected in \( G \). The minimum \( k \) for which there is a connected resolving \( k \)-partition of \( V(G) \) is called the connected partition dimension of \( G \), denoted by \( cpd(G) \). In this paper, the connected partition dimension of the unicyclic graphs is calculated and bounds are proposed.
Let \( G = (V, E) \) be a finite graph, where \( V(G) \) and \( E(G) \) are the (non-empty) sets of vertices and edges of \( G \). An \((a, d)\)-\({edge-antimagic\; total\; labeling}\) is a bijection \( \beta \) from \( V(G) \cup E(G) \) to the set of consecutive integers \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) with the property that the set of all the edge-weights, \( w(uv) = \beta(u) + \beta(uv) + \beta(v) \), for \( uv \in E(G) \), is \( \{a, a + d, a + 2d, \dots, a + (|E(G)| – 1)d\} \), for two fixed integers \( a > 0 \) and \( d \geq 0 \). Such a labeling is super if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings for disjoint unions of multiple copies of a regular caterpillar.
The term mode graph was introduced by Boland, Kaufman, and Panrong to define a connected graph \( G \) such that, for every pair of vertices \( v, w \) in \( G \), the number of vertices with eccentricity \( e(v) \) is equal to the number of vertices with eccentricity \( e(w) \). As a natural extension to this work, the concept of an antimode graph was introduced to describe a graph for which, if \( e(v) \neq e(w) \), then the number of vertices with eccentricity \( e(v) \) is not equal to the number of vertices with eccentricity \( e(w) \). In this paper, we determine the existence of some classes of antimode graphs, namely equisequential and \((a, d)\)-antimode graphs.
By an \((a, d)\)-edge-antimagic total labeling of a graph \( G(V, E) \), we mean a bijective function \( f \) from \( V(G) \cup E(G) \) onto the set \( \{1, 2, \dots, |V(G)| + |E(G)|\} \) such that the set of all the edge-weights, \( w(uv) = f(u) + f(uv) + f(v) \), for \( uv \in E(G) \), is \( \{a, a+d, a+2d, \dots, a + (|E(G)| – 1)d\} \), for two integers \( a > 0 \) and \( d \geq 0 \).
In this paper, we study the edge-antimagic properties for the disjoint union of complete \( s \)-partite graphs.
We study the number of super edge-magic (bipartite) graphs from an asymptotic point of view.
It is well known that apart from the Petersen graph, there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and \(\delta\) vertices less than the Moore bound, where \(\delta\) is odd. Additionally, it is known that there exist only three graphs of maximum degree 3 and 2 vertices less than the Moore bound. In this paper, we consider graphs of maximum degree 3, diameter \( D \geq 2 \), and 4 vertices less than the Moore bound, denoted as \((3, D, 4)\)-graphs. We obtain all non-isomorphic \((3, D, 4)\)-graphs for \( D = 2 \). Furthermore, for any diameter \( D \), we consider the girth of \((3, D, 4)\)-graphs. By a counting argument, it is easy to see that the girth is at least \( 2D – 2 \). The main contribution of this paper is that we prove that the girth of a \((3, D, 4)\)-graph is at least \( 2D – 1 \). Finally, for \( D > 4 \), we conjecture that the girth of a \((3, D, 4)\)-graph is \( 2D \).
A fast direct method for obtaining the incidence matrix of a finite projective plane of order \( n \) via \( n-1 \) mutually orthogonal \( n \times n \) Latin squares is described. Conversely, \( n-1 \) mutually orthogonal \( n \times n \) Latin squares are directly exhibited from the incidence matrix of a projective plane of order \( n \). A projective plane of order \( n \) can also be described via a digraph complete set of Latin squares, and a new procedure for doing this will also be described.
Let \(K_v\) be a complete graph with \(v\) vertices, and \(G = (V(G), E(G))\) be a finite simple graph. A \(G\)-design \(G-GD_\lambda(v)\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). In this paper, the existence of graph designs \(G-GD_\lambda(v)\), \(\lambda > 1\), for eight graphs \(G\) with six vertices and eight edges is completely solved.
A \({weighted \;graph}\) is one in which every edge \(e\) is assigned a nonnegative number \(w(e)\), called the \({weight}\) of \(e\). The \({weight\; of \;a \;cycle}\) is defined as the sum of the weights of its edges. The \({weighted \;degree}\) of a vertex is the sum of the weights of the edges incident with it. In this paper, motivated by a recent result of Fujisawa, we prove that a \(2\)-connected weighted graph \(G\) contains either a Hamilton cycle or a cycle of weight at least \(2m/3\) if it satisfies the following conditions:
\((1)\) The weighted degree sum of every three pairwise nonadjacent vertices is at least \(m\);\((2)\)In each induced claw and each induced modified claw of \(G\), all edges have the same weight.This extends a theorem of Zhang, Broersma and Li.
The \({restricted edge-connectivity}\) of a graph is an important parameter to measure fault-tolerance of interconnection networks. This paper determines that the restricted edge-connectivity of the de Bruijn digraph \(B(d,n)\) is equal to \(2d – 2\) for \(d \geq 2\) and \(n \geq 2\) except \(B(2,2)\). As consequences, the super edge-connectedness of \(B(d,n)\) is obtained immediately.
An edge coloring of a graph is called \({square-free}\) if the sequence of colors on certain walks is not a square, that is not of the form \(x_1, \ldots, x_m, x_{1}, \ldots, x_m\) for any \(m \in \mathbb{N}\). Recently, various classes of walks have been suggested to be considered in the above definition. We construct graphs, for which the minimum number of colors needed for a square-free coloring is different if the considered set of walks vary, solving a problem posed by Brešar and Klavžar. We also prove the following: if an edge coloring of \(G\) is not square-free (even in the most general sense), then the length of the shortest square walk is at most \(8|E(G)|^2\). Hence, the necessary number of colors for a square-free coloring is algorithmically computable.
If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+ (x)\) and \(d^- (x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+ (x),d^- (x)\} – \min\{d^+ (y), d^- (y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)).
A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. Recently, Volkmann and Winzen \([9]\) proved that \(c\)-partite tournaments with \(i_g(D) = 1\) and \(c \geq 3\) or \(i_g(D) = 2\) and \(c \geq 5\) contain a Hamiltonian path. Furthermore, they showed that these bounds are best possible.
Now, it is a natural question to generalize this problem by asking for the minimal value \(g(i,k)\) with \(i,k \geq 1\) arbitrary such that all \(c\)-partite tournaments \(D\) with \(i_g(D) \leq i\) and \(c \geq g(i,k)\) have a path covering number \(pc(D) \leq k\). In this paper, we will prove that \(4i-4k \leq g(i,k) \leq 4i-3k-1\), when \(i \geq k+2\). Especially in the case \(k = 1\), this yields that \(g(i, 1) = 4i-4\), which means that all \(c\)-partite tournaments \(D\) with the global irregularity \(i_g(D) = i\) and \(c \geq 4i-4\) contain a Hamiltonian path.
In this paper, we discuss a problem on packing a unit cube with smaller cubes, which is a generalization of one of Erdős’ favorite problems: the square-packing problem. We first give the definition of the packing function \(f_3(n)\), then give the bounds for \(f_3(n)\).
A set \(S\) of vertices in a graph \(G = (V, E)\) is a restrained dominating set of \(G\) if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V \setminus S\). The graph \(G\) is called restrained domination excellent if every vertex belongs to some minimum restrained dominating set of \(G\). We provide a characterization of restrained domination excellent trees.
In this paper, \(q\)-analogues of the Pascal matrix and the symmetric Pascal matrix are studied. It is shown that the \(q\)-Pascal matrix \(\mathcal{P}_n\) can be factorized by special matrices and the symmetric \(q\)-Pascal matrix \(\mathcal{Q}_n\) has the LDU-factorization and the Cholesky factorization. As byproducts, some \(q\)-binomial identities are produced by linear algebra. Furthermore, these matrices are generalized in one or two variables, where a short formula for all powers of \(q\)-Pascal functional matrix \(\mathcal{P}_n[x]\) is given. Finally, it is similar to Pascal functional matrix, we have the exponential form for \(q\)-Pascal functional matrix.
We view a lobster in this paper as below. A lobster with diameter at least five has a unique path \(H = x_0, x_1, \ldots, x_m\) with the property that, besides the adjacencies in \(H\), both \(x_0\) and \(x_m\) are adjacent to the centers of at least one \(K_{i,s}\), where \(s > 0\), and each \(x_i\), \(1 \leq i \leq m-1\), is at most adjacent to the centers of some \(K_{1,s}\), where \(s \geq 0\). This unique path \(H\) is called the central path of the lobster. We call \(K_{1,s}\) an even branch if \(s\) is nonzero even, an odd branch if \(s\) is odd, and a pendant branch if \(s = 0\). In this paper, we give graceful labelings to some new classes of lobsters with diameter at least five. In these lobsters, the degree of each vertex \(x_i\), \(0 \leq i \leq m-1\), is even and the degree of \(x_m\) may be odd or even, and we have one of the following features:
In this paper, an algorithm for constructing self-centered graphs from trees and two more algorithms for constructing self-centered graphs from a given connected graph \(G\), by adding edges are discussed. Motivated by this, a new graph theoretic parameter \(sc_r(G)\), the minimum number of edges added to form a self-centered graph from \(G\) is defined. Bounds for this parameter are obtained and exact values of this parameter for several classes of graphs are also obtained.
A \((k;g)\)-graph is a \(k\)-regular graph with girth \(g\). A \((k;g)\)-cage is a \((k;g)\)-graph with the least number of vertices. In this note, we show that a \((k;g)\)-cage has an \(r\)-factor of girth at least \(g\) containing or avoiding a given edge for all \(r\), \(1 \leq r \leq k-1\).
This paper deals with the problem of constructing Hamiltonian paths of optimal weights in Halin graphs. There are three versions of the Hamiltonian path: none or one or two of end-vertices are specified. We present \(O(|V|)\) algorithms for all the versions of the problem.
It is widely recognized that certain graph-theoretic extremal questions play a major role in the study of communication network vulnerability. These extremal problems are special cases of questions concerning the realizability of graph invariants. We define a CS\((p, q, \lambda, \delta)\) graph as a connected, separable graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). In this notation, if the “CS” is omitted the graph is not necessarily connected and separable. An arbitrary quadruple of integers \((a, b, c, d)\) is called CS\((p, q, \lambda, \delta)\) realizable if there is a CS\((p, q, \lambda, \delta)\) graph with \(p = a\), \(q = b\), \(\lambda = c\) and \(\delta = d\). Necessary and sufficient conditions for a quadruple to be CS\((p, q, \lambda, \delta)\) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta)\), \((p, q, \lambda,\Delta )\), \((p, q, \delta, \Delta)\), \((p, q, \lambda, \delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(\Delta\) denotes the maximum degree for all points in a graph and \(\kappa\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa)\), \((p, q, \lambda)\), \((p, q, \delta)\), \((p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.
An incidence graph of a given graph \(G\), denoted by \(I(G)\), has its own vertex set \(V(I(G)) = \{(ve) | v \in V(G), e \in E(G) \text{ and } v \text{ is incident to } e \text{ in } G\}\) such that the pair \(((ue)(vf))\) of vertices \((ue) (vf) \in V(I(G))\) is an edge of \(I(G)\) if and only if there exists at least one case of \(u = v, e = f, uv = e\) or \(uv = f\). In this paper, we carry out a constructive definition on incidence graphs, and investigate some properties of incidence graphs and some edge-colorings on several classes of them.
The maximum possible toughness among graphs with \(n\) vertices and \(m\) edges is considered for \(m \geq \lceil n^2/4 \rceil\). We thus extend results known for \(m \geq n\lfloor n/3 \rfloor\). When \(n\) is even, all of the values are determined. When \(n\) is odd, some values are determined, and the difficulties are discussed, leaving open questions.
In this paper, we, by means of Rosa’s \(\alpha\)-labelling and \(k\)-graceful labelling, prove that generalized spiders, generalized caterpillars, and generalized path-block chains are graceful under some conditions. Some of the results are stronger than that obtained in \([4]\).
We study convexity with respect to a definition of fractional independence in a graph \(G\) that is quantified over neighbourhoods rather than edges. The graphs that admit a so-called universal maximal fractional independent set are characterized, as are all such sets. A characterization is given of the maximal fractional independent sets which cannot be obtained as a proper convex combination of two other such sets.
In this paper, we consider the relationship between the toughness and the existence of fractional \(f\)-factors. It is proved that a graph $G$ has a fractional \(f\)-factor if \(t(G) \geq \frac{b^2+b}{a}-\frac{b+1}{b}\). Furthermore, we show that the result is best possible in some sense.
It is always fascinating to see what results when seemingly different areas of mathematics overlap. This article reveals one such result; number theory and linear algebra are intertwined to yield complex factorizations of the classic Fibonacci, Pell, Jacobsthal, and Mersenne numbers. Also, in this paper we define a new matrix generalization of the Fibonacci numbers, and using essentially a matrix approach we show some properties of this matrix sequence.
The notion of meandric polygons is introduced in this paper. A bijection exists between the set of meandric polygons and that of closed meanders. We use these polygons to enumerate the set of meanders which have a fixed number of arcs of the meandric curves lying above and below the horizontal line at a given point.
In this paper, we give a sufficient and necessary condition for a \(k\)-extendable graph to be \(2k\)-factor-critical when \(k = \frac{v}{4}\), and prove some results on independence numbers in \(n\)-factor-critical graphs and \(k\frac{1}{2}\)-extendable graphs.
In this paper, we show that some families of graphs are arbitrarily graceful or almost graceful.
We consider the lattice of order ideals of the union of an \(n\)-element fence and an antichain of size \(i\), whose Hasse diagram turns out to be isomorphic to the \(i\)-th extended Fibonacci cube. We prove that the Whitney numbers of these lattices form a unimodal sequence satisfying a particular property, called \({alternating}\), we find the maximum level of the game sequence and determine the exact values of these numbers.
Let \(D\) be a strongly connected digraph with order at least two. Let \(T(D)\) denote the total digraph of \(D\), and let \(\kappa(D)\) and \(\lambda(D)\) denote the connectivity and arc-connectivity of \(D\), respectively. In this paper, we study super-arc-connected and super-connected total digraphs. The following results are obtained:
A vertex\(|\)matching-partition \((V|M)\) of a simple graph \(G\) is a spanning collection of vertices and independent edges of \(G\). Let vertex \(v \in V\) have weight \(w_v\) and edge \(e \in M\) have weight \(w_e\). Then the weight of \(V|M\) is \(w(V|M) = \prod_{v \in V} w_v + \prod_{e \in M} w_e\). Define the vertex|matching-partition function of \(G\) as \(W(G) = \sum_{V|M} w(V|M)\).
In this paper, we study this function when \(G\) is a path and a cycle. We generate all orthogonal polynomials as vertex|matching-partition functions of suitably labelled paths, and indicate how to find their derivatives in some cases. Here Taylor’s Expansion is used, and an application to associated polynomials is given. We also give a combinatorial interpretation of coefficients in the case of multiplicative and additive weights. Results are extended to the weighted cycle.
Let \(k\) be a nonnegative integer, and let \(\gamma(G)\) and \(i(G)\) denote the domination number and the independent domination number of a graph \(G\), respectively. The so-called \(i_k\)-perfect graphs consist of all such graphs \(G\) in which \(i(H) – \gamma(H) \leq k\) holds for every induced subgraph \(H\) of \(G\). This concept, introduced by I. Zverovich in \([5]\), generalizes the well-known domination perfect graphs. He conjectured that \(i\gamma (k)\)-perfect graphs also have a finite forbidden induced subgraphs characterization, as is the case for domination perfect graphs. Recently, Dohmen, Rautenbach, and Volkmann obtained such a characterization for all \(i\gamma(1)\)-perfect forests. In this paper, we characterize the \(i\gamma(1)\)-perfect graphs with girth at least six.
Let \(G\) be a simple and connected graph of order \(p \geq 2\). A \({proper k-total-coloring}\) of a graph \(G\) is a mapping \(f\) from \(V(G) \bigcup E(G)\) into \(\{1, 2, \ldots, k\}\) such that every two adjacent or incident elements of \(V(G) \bigcup E(G)\) are assigned different colors. Let \(C_f(u) = f(u) \bigcup \{f(uv) | uv \in E(G)\}\) be the \({neighbor \;color-set}\) of \(u\). If \(C_f(u) \neq C_f(v)\) for any two vertices \(u\) and \(v\) of \(V(G)\), we say \(f\) is a \({vertex-distinguishing \;proper\; k-total-coloring}\) of \(G\), or a \({k-VDT-coloring}\) of \(G\) for short. The minimal number of all \(k\)-VDT-colorings of \(G\) is denoted by \(\chi_{vt}(G)\), and it is called the \({VDTC \;chromatic \;number}\) of \(G\). For some special families of graphs, such as the complete graph \(K_n\), complete bipartite graph \(K_{m,n}\), path \(P_m\), and circle \(C_m\), etc., we determine their VDTC chromatic numbers and propose a conjecture in this article.
The cochromatic number of a graph \(G\), denoted by \(z(G)\), is the fewest number of parts we need to partition \(V(G)\) so that each part induces in \(G\) an empty or a complete graph. A graph \(G\) with \(z(G) = n\) is called \({critically n-cochromatic}\) if \(z(G – v) = n – 1\) for each vertex \(v\) of \(G\), and \({minimally n-cochromatic}\) if \(z(G – e) = n – 1\) for each edge \(e\) of \(G\).
We show that for a graph \(G\), \(K_{1} \cup G \cup K_{2} \cup \cdots \cup K_{n-1} \cup G\) is a critically \(n\)-cochromatic graph if and only if \(G\) is \(K_{n}\), \((n \geq 2)\). We consider general minimally cochromatic graphs and obtain a result that a minimally cochromatic graph is either a critically cochromatic graph or a critically cochromatic graph plus some isolated vertices. We also prove that given a graph \(G\), then \(K_{1} \cup G \cup K_{2} \cup \cdots \cup K_{n-1} \cup G\) \((n \geq 2)\) is minimally \(n\)-cochromatic if and only if \(G\) is \(K_{n}\) or \(\overline{K_{n-1}} \cup \overline{K_{p}}\) for \(p \geq 1\). We close by giving some properties of minimally \(n\)-cochromatic graphs.
We examine the inverse domination number of a graph, as well as two reasonable candidates for the fractional analogue of this parameter. We also examine the relations among these and other graph parameters. In particular, we show that both proposed fractional analogues of the inverse domination number are no greater than the fractional independence number. These results establish the fractional analogue of a well-known conjecture about the inverse domination and vertex independence numbers of a graph.
We consider a \(2\)-coloring of arcs on the primitive extremal tournament with the largest exponent on \(n\) vertices and \(m\) arcs. This \(2\)-colored digraph is a \(2\)-primitive tournament. Then we consider the \(2\)-exponent of a \(2\)-primitive tournament. In this paper, we give an upper bound for the \(2\)-exponent of the primitive extremal tournament.
The Fibonacci graph \( G_n \) is the graph whose vertex set is the collection of \( n \)-bit binary strings having no contiguous ones, and two vertices are adjacent if and only if their Hamming distance is one. Values of several graphical invariants are determined for these graphs, and bounds are found for other invariants.
Given a configuration of pebbles on the vertices of a connected graph \( G \), a \({pebbling\; move}\) is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. We introduce the notion of domination cover pebbling, obtained by combining graph cover pebbling with the theory of domination in graphs. The domination cover pebbling number, \( \psi(G) \), of a graph \( G \) is the minimum number of pebbles that are placed on \( V(G) \) such that after a sequence of pebbling moves, the set of vertices with pebbles forms a dominating set of \( G \), regardless of the initial configuration of pebbles. We discuss basic results and determine \( \psi(G) \) for paths, cycles, and complete binary trees.
We show that the double domination number of an \( n \)-vertex, isolate-free graph with minimum degree \( \delta \) is bounded above by \(\frac{n(\ln(\delta + 1) + \ln \delta + 1)}{\delta}.\) This result improves a previous bound obtained by J. Harant and M. A. Henning [On double domination in graphs, \({Discuss. Math. Graph Theory}\) \({25} (2005), 29-34]\). Further, we show that for fixed \( k \) and large \( \delta \), the \( k \)-tuple domination number is at most \(\frac{n(\ln \delta + (k – 1 + o(1))\ln \ln \delta)}{\delta},\) a bound that is essentially best possible.
Let \(\alpha\)-resolvable STS(\(v\)) denote a Steiner triple system of order \(v\) whose blocks are partitioned into classes such that each point of the design occurs in precisely \(\alpha\) blocks in each class. We show that for \(v \equiv u \equiv 1 \pmod{6}\) and \(v \geq 3u + 4\), there exists an \(\alpha\)-resolvable STS(\(v\)) containing an \(\alpha\)-resolvable sub-STS(\(u\)) for all suitable \(\alpha\).
A vertex set \( S \subseteq V(G) \) of a graph \( G \) is a \( 2 \)-dominating set of \( G \) if \( |N(v) \cap S| \geq 2 \) for every vertex \( v \in (V(G) – S) \), where \( N(v) \) is the neighborhood of \( v \). The \( 2 \)-domination number \( \gamma_2(G) \) of graph \( G \) is the minimum cardinality among the \( 2 \)-dominating sets of \( G \). In this paper, we present the following Nordhaus-Gaddum-type result for the \( 2 \)-domination number. If \( G \) is a graph of order \( n \), and \( \bar{G} \) is the complement of \( G \), then
\[ \gamma_2(G) + \gamma_2(\bar{G}) \leq n + 2, \]
and this bound is best possible in some sense.
The Graph Isomorphism (GI) problem asks if two graphs are isomorphic. Algorithms which solve GI have applications in, but not limited to, SAT solver engines, isomorph-free generation, combinatorial analysis, and analyzing chemical structures. However, no algorithm has been found which solves GI in polynomial time, implying that hard instances should exist. One of the most popular algorithms, implemented in the software package nauty, canonically labels a graph and outputs generators for its automorphism group. In this paper, we present some methods that improve its performance on graphs that are known to pose difficulty.
Let \( C \) be the set of distinct ways in which the vertices of a \( 5 \)-cycle may be coloured with at most two colours, called \({colouring\; types}\), and let \( S \subseteq C \). Suppose we colour the vertices of \( K_v \) with at most two colours. If \( \mathcal{D} \) is a \( 5 \)-cycle decomposition of \( K_v \), such that the colouring type of each \( 5 \)-cycle is in \( S \), and every colouring type in \( S \) is represented in \( \mathcal{D} \), then \( \mathcal{D} \) is said to have a \emph{proper colouring type} \( S \). For all \( S \) with \( |S| \leq 2 \), we determine some necessary conditions for the existence of a \( 5 \)-cycle decomposition of \( K_v \) with proper colouring type \( S \). In many cases, we show that these conditions are also sufficient.
Most computer algebra packages for Weyl groups use generators and relations and the Weyl group elements are expressed as reduced words in the generators. This representation is not unique and leads to computational problems. In [HHR06], the authors introduce the representation of Weyl group elements uniquely as signed permutations. This representation is useful for the study of symmetric spaces and their representations.
A computer algebra package enabling one to do computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this paper, we use the representation of Weyl group elements as signed permutations to improve the algorithms of [DH05]. These algorithms compute the fine structure of symmetric spaces and nice bases for local symmetric spaces.
A vertex subset \( X \) of a simple graph is called OC-irredundant (respectively CO-irredundant) if for each \( v \in X \), \( N(v) – N[X – \{v\}] \neq \emptyset \) (respectively \( N[v] – N(X – \{v\}) \neq \emptyset \)). Sharp bounds involving order and maximum degree for the minimum cardinality of a maximal OC-irredundant set and a maximal CO-irredundant set of a tree are obtained, and extremal trees are exhibited.
For any \( k \in \mathds{N} \), a graph \( G = (V, E) \) is said to be \( \mathds{Z}_k \)-magic if there exists a labeling \( l: E(G) \to \mathds{Z}_k – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathds{Z}_k \), defined by
\[ l^+(v) = \sum_{uv \in E(G)} l(uv) \]
is a constant map. For a given graph \( G \), the set of all \( k \in \mathds{N} \) for which \( G \) is \( \mathds{Z}_k \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will consider the functional extensions of \( P_n \) (\( n = 2, 3, 4 \)) and will determine their integer-magic spectra.
Let \( G(V, E) \) be a weighted connected simple graph. Given a pair of vertices \( u, v \in V \), let \( Max(u, v) \) denote the maximum average path value over all simple paths from \( u \) to \( v \). For a given simple path from \( u \) to \( v \), the average path value, \( apv_P(u, v) = \frac{min(P)}{length(P)} \), where \( min(P) \) is the weight of the minimum weight edge in the path \( P \) and \( length(P) \) is the number of edges in \( P \). This notion of average path value has been used in the analysis of social networks. Algorithms are presented for the calculation of \emph{average path value}.
For the purpose of large scale computing, we are interested in linking computers into large interconnection networks. In order for these networks to be useful, the underlying graph must possess desirable properties such as a large number of vertices, high connectivity, and small diameter. In this paper, we are interested in the alternating group graph, as an interconnection network, and the \( k \)-Disjoint Path Problem. In 2005, Cheng, Kikas, and Kruk showed that the alternating group graph, \( AG_n \), has the \( (n – 2) \)-Disjoint Path Property. However, their proof was an existence proof only; they did not show how to actually construct the \( (n – 2) \) disjoint paths. In 2006, Boats, Kikas, and Oleksik developed an algorithm for constructing three disjoint paths in the graph \( AG_5 \). Their algorithm exploited the hierarchical structure of \( AG_n \) to construct the paths. In this paper, we develop a purely algebraic algorithm that constructs the \( (n – 2) \) disjoint paths from scratch. This algebraic approach can be used for other Cayley graphs such as the split-star and the star graphs. Indeed, we believe that our approach can be used for any Cayley graph. We close with remarks on possible research directions stemming from this work.
The computation of the maximum toughness among graphs with \( n \) vertices and \( m \) edges is considered for \( \lceil{5n}/{2}\rceil \leq m < 3n \). We show that there are only finitely many cases in which the toughness value \( \frac{5}{2} \) cannot be achieved. This is in stark contrast with the known result that there is a \( \frac{3}{2} \)-tough graph on \( n \) vertices and \( \lceil{3n}/{2}\rceil \) edges if and only if \( n \equiv 0, 5 \pmod{6} \). However, constructions related to those used in the cubic case are also employed here. Our constructions additionally provide an infinite family of graphs that are supertough and not \( K_{1,3} \)-free.
A simple graphoidal cover of a graph \( G \) is a collection \( \psi \) of (not necessarily open) paths in \( G \) such that every path in \( \psi \) has at least two vertices, every vertex of \( G \) is an internal vertex of at most one path in \( \psi \), every edge of \( G \) is in exactly one path in \( \psi \), and any two paths in \( \psi \) have at most one vertex in common. The minimum cardinality of a simple graphoidal cover of \( G \) is called the simple graphoidal covering number of \( G \) and is denoted by \( \eta_s(G) \). In this paper, we determine the value of \( \eta_s \) for several families of graphs. We also obtain several bounds for \( \eta_s \) and characterize graphs attaining the bounds.
In this paper, we discuss how the addition of a new edge affects the irregularity strength of a graph.
Let \( G \) be a graph of size \( n \) with vertex set \( V(G) \) and edge set \( E(G) \). A \( \sigma \)-labeling of \( G \) is a one-to-one function \( f: V(G) \to \{0, 1, \dots, 2n\} \) such that \( \{|f(u) – f(v)| : \{u, v\} \in E(G)\} = \{1, 2, \dots, n\} \). Such a labeling of \( G \) yields cyclic \( G \)-decompositions of \( K_{2n+1} \) and of \( K_{2n+2} – F \), where \( F \) is a \( 1 \)-factor of \( K_{2n+2} \). It is conjectured that a \( 2 \)-regular graph of size \( n \) has a \( \sigma \)-labeling if and only if \( n \equiv 0 \) or \( 3 \pmod{4} \). We show that this conjecture holds when the graph has at most three components.
The Kneser graph \( K(m, n) \) (when \( m > 2n \)) has the \( n \)-subsets of an \( m \)-set as its vertices, two vertices being adjacent in \( K(m, n) \) whenever they are disjoint sets. The \( k \)th-chromatic number of any graph \( G \) (denoted by \( \chi_k(G) \)) is the least integer \( t \) such that the vertices can be assigned \( k \)-subsets of \( \{1, 2, \dots, t\} \) with adjacent vertices receiving disjoint \( k \)-sets. S. Stahl has conjectured that, if \( k = qn – r \) where \( q \geq 1 \) and \( 0 \leq r < n \), then \( \chi_k(K(m, n)) = qm – 2r \). This expression is easily verified when \( r = 0 \); Stahl has also established its validity for \( q = 1 \), for \( m = 2n + 1 \) and for \( k = 2, 3 \). We show here that the expression is also valid for all \( q \geq 2 \) in the following further classes of cases:
We prove that for a finite planar space \( S = (\mathcal{P}, \mathcal{L}, \mathcal{H}) \) with no disjoint planes and with a constant number of planes on a line, the number \( \ell \) of lines is greater than or equal to the number \( c \) of planes, and the equality holds true if and only if \( S \) is either the finite Desarguesian 4-dimensional projective space \( PG(4,q) \), or the complete graph \( K_5 \).
A \((p, q)\)-graph \( G \) is said to be \(\textbf{edge graceful}\) if the edges can be labeled by \( 1, 2, \ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. J. Mitchem and A. Simoson introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for some classes of graphs.
A graph \( G = (V, E) \) of order \( p \) and size \( q \) is said to be \(\textbf{super edge-graceful}\) (SEG) if there exists a bijection
\[
\text{f: E} \to
\begin{cases}
\{0, +1, -1, +2, -2, \ldots, \frac{q-1}{2}, -\frac{q-1}{2}\} & \text{if } q \text{ is odd} \\
\{+1, -1, +2, -2, \ldots, \frac{q}{2}, -\frac{q}{2}\} & \text{if } q \text{ is even}
\end{cases}
\]
such that the induced vertex labeling \( f^* \) defined by \( f^*(u) = \sum \{ f(u, v) : (u, v) \in E \} \) has the property:
\[
f^*: V \to
\begin{cases}
\{0, +1, -1, \ldots, +\frac{p-1}{2}, -\frac{p-1}{2}\} & \text{if } p \text{ is odd} \\
\{+1, -1, \ldots, +\frac{p}{2}, -\frac{p}{2}\} & \text{if } p \text{ is even}
\end{cases}
\]
is a bijection.
The conjecture is still unsettled. In this paper, we first characterize spiders of even orders which are not SEG. We then exhibit some spiders of even orders which are SEG of diameter at most four. By the concepts of the irreducible part of an even tree \( T \), we show that an infinite number of spiders of even orders are SEG. Finally, we provide some conjectures for further research.
Let \(G = (V(G), E(G))\) be a nonempty graph (may have parallel edges). The line graph \(L(G)\) of \(G\) is the graph with \(V(L(G)) = E(G)\), and in which two vertices \(e\) and \(e’\) are joined by an edge if and only if they have a common vertex in \(G\). We call the complement of \(L(G)\) as the jump graph. In this note, we give a simple sufficient and necessary condition for a jump graph to have a perfect matching.
We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not yield designs with new parameters, but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.
A shell graph of order \(n\), denoted by \(H(n, n-3)\), is the graph obtained from the cycle \(C_n\) of order \(n\) by adding \(n-3\) chords incident with a common vertex, say \(u\). Let \(v\) be a vertex adjacent to \(u\) in \(C_n\). Sethuraman and Selvaraju \([3]\) conjectured that for all \(k \geq 1\) and for all \(n_i \geq 4\), \(1 \leq i \leq k\), one edge \((uv)\) union of \(k\)-shell graphs \(H(n_i, n_i – 3)\) is cordial. In this paper, we settle this conjecture affirmatively.
In this paper, we give formulas for the sums of generalized order-\(k\) Fibonacci, Pell, and similar other sequences, which we obtain using matrix methods. As applications, we give explicit formulas for the Tribonacci and Tetranacci numbers.
A \((g, f)\)-coloring is a generalized edge-coloring in which each color appears at each vertex \(v\) at least \(g(v)\) and at most \(f(v)\) times, where \(g(v)\) and \(f(v)\) are nonnegative and positive integers assigned to each vertex \(v\), respectively. The minimum number of colors used by a \((g, f)\)-coloring of \(G\) is called the \((g, f)\)-chromatic index of \(G\). The maximum number of colors used by a \((g, f)\)-coloring of \(G\) is called the upper \((g, f)\)-chromatic index of \(G\). In this paper, we determine the \((g, f)\)-chromatic index and the upper \((g, f)\)-chromatic index in some cases.
The Szeged index extends the Wiener index for cyclic graphs by counting the number of atoms on both sides of each bond and summing these counts. This index was introduced by Ivan Gutman at the Attila Jozsef University in Szeged in \(1994\), and is thus called the Szeged index. In this paper, we introduce a novel method for enumerating by cuts. Using this method, an exact formula for the Szeged index of a zig-zag polyhex nanotube \(T = TUHC_6{[p,q]}\) is computed for the first time.
In this study, we showed that an \((n+1)\)-regular linear space, which is the complement of a linear space having points not on \(m+1\) lines such that no three are concurrent in a projective subplane of odd order \(m\), \(m \geq 9\), could be embedded into a projective plane of order \(n\) as the complement of Ostrom’s hyperbolic plane.
For general graphs \(G\), it is known \([6]\) that the minimal length of an addressing scheme, denoted by \(N(G)\), is less than or equal to \(|G| – 1\). In this paper, we prove that for almost all complete bipartite graphs \(K_{m,n}\), \(N(K_{m,n}) = |K_{m,n}| – 2\).
A vertex subversion strategy of a graph \(G\) is a set of vertices \(X \subseteq V(G)\) whose closed neighborhood is deleted from \(G\). The survival subgraph is denoted by \(G/X\). The vertex-neighbor-integrity of \(G\) is defined to be \(VNI(G) = \min\{|X| + r(G/X) : X \subseteq V(G)\},\) where \(r(G/X)\) is the order of a largest component in \(G/X\). This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. It was proved by Gambrell that the decision problem of computing the vertex-neighbor-integrity of a graph is NP-complete. In this paper, we evaluate the vertex-neighbor-integrity of the composition graph of two paths.
In this paper, we prove that a matroid with at least two elements is connected if and only if it can be obtained from a loop by a nonempty sequence of non-trivial single-element extensions and series extensions.
Let \(G\) and \(H\) be graphs with a common vertex set \(V\), such that \(G – i \cong H – i\)for all \(i \in V\). Let \(p_i\) be the permutation of \(V – i\) that maps \(G – i\) to \(H – i\), and let \(q_i\) denote the permutation obtained from \(p_i\) by mapping \(i\) to \(i\). It is shown that certain algebraic relations involving the edges of \(G\) and the permutations \(q_iq_j^{-1}\) and \(q_iq_k^{-1}\), where \(i, j, k \in V\) are distinct vertices, often force \(G\) and \(H\) to be isomorphic.
The factorization of matrix \(A\) with entries \(a_{i,j}\) determined by \(a_{i,j} = \alpha a_{i-1,j-1} + \beta a_{i,j-1}\) is derived as \(A = TP^T\). An interesting factorization of matrix \(B\) with entries \(b_{i,j} = \alpha b_{i-1,j} + \beta b_{i,j-1}\) is given by \(B = P[\alpha]TP^{T}[\beta]\). The beautiful factorization of matrix \(C\) whose entries satisfy \(c_{i,j} = \alpha c_{i-1,j} + \beta c_{i-1,j-1} + Ye_{i-1,j-1}\) is founded to be \(C = P[\alpha]DP^T[\beta]\), where \(T\) is a Toeplitz matrix, and \(P\) and \(P[\alpha]\) are Pascal matrices. The matrix product factorization to the problem is solved perfectly so far.
Dirac showed that a \(2\)-connected graph of order \(n\) with minimum degree \(\delta\) has circumference at least \(\min\{2\delta, n\}\). We prove that a \(2\)-connected, triangle-free graph \(G\) of order \(n\) with minimum degree \(\delta\) either has circumference at least \(\min\{4\delta – 4, n\}\), or every longest cycle in \(G\) is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree \(\delta\) whose longest cycles have length \(4\delta – 4\), and are not dominating.
The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. It is proved here that \(\lceil \frac{\omega(G)}{2}\rceil \leq vla(G) \leq \lceil \frac{\omega(G)+1}{2}\rceil\) for a claw-free connected graph \(G\) having \(\Delta(G) \leq 6\), where \(\omega(G)\) is the clique number of \(G\).
An \(f\)-coloring of a graph \(G\) is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\) and denoted by \(\chi’_f(G)\). Any simple graph \(G\) has the \(f\)-chromatic index equal to \(\Delta_f(G)\) or \(\Delta_f(G) + 1\), where \(\Delta_f(G) = \max_{v \in V}\{\lceil \frac{d(v)} {f(v)}\rceil\}\). If \(\chi’_f(G) = \Delta_f(G)\), then \(G\) is of \(C_f\) \(1\); otherwise \(G\) is of \(C_f\) \(2\). In this paper, two sufficient conditions for a regular graph to be of \(C_f\) \(1\) or \(C_f\) \(2\) are obtained and two necessary and sufficient conditions for a regular graph to be of \(C_f\) \(1\) are also presented.
Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f: V(G) \to A\) induces an edge labeling \(f^*: E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \text{card}\{v \in V(G): f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E(G): f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i,j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A-friendly\) if \(|v_f(i) – v_f(j)| \leq 1\) for all \((i,j) \in A \times A\). If \(c(f)\) is a \((0,1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the \({friendly index set}\) of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)|: \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In this paper, we determine the friendly index set of cycles, complete graphs, and some bipartite graphs.
In this paper, several constructions are presented for balanced incomplete block designs with nested rows and columns. Some of them refine theorems due to Hishida and Jimbo \([6]\) and Uddin and Morgan \([17]\), and some of them give parameters which have not been available before.
A vertex-distinguishing edge-coloring (VDEC) of a simple graph \(G\) which contains no more than one isolated vertex and no isolated edge is equitable (VDEEC) if the absolute value of the difference between the number of edges colored by color \(i\) and the number of edges colored by color \(j\) is at most one. The minimal number of colors needed such that \(G\) has a VDEEC is called the vertex-distinguishing equitable chromatic index of \(G\). In this paper, we propose two conjectures after investigating VDEECs on some special families of graphs, such as the stars, fans, wheels, complete graphs, complete bipartite graphs, etc.
The eccentricity \(e(v)\) of a vertex \(v\) in a strongly connected digraph \(G\) is the maximum distance from \(v\). The eccentricity sequence of a digraph is the list of eccentricities of its vertices given in non-decreasing order. A sequence of positive integers is a digraphical eccentric sequence if it is the eccentricity sequence of some digraph. A set of positive integers \(S\) is a digraphical eccentric set if there is a digraph \(G\) such that \(S = \{e(v), v \in V(G)\}\). In this paper, we present some necessary and sufficient conditions for a sequence \(S\) to be a digraphical eccentric sequence. In some particular cases, where either the minimum or the maximum value of \(S\) is fixed, a characterization is derived. We also characterize digraphical eccentric sets.
Let \(C_m\) be a cycle on \(m (\geq 3)\) vertices and let \(\ominus_{n-m}C_m\) denote the class of graphs obtained from \(C_m\) by adding \(n-m (\geq 1)\) distinct pendent edges to the vertices of \(C_m\). In this paper, it is proved that for every \(T\) in \(\ominus_{n-m}C_m\), the complete graph \(K_{2n+1}\) can be cyclically decomposed into the isomorphic copies of \(T\). Moreover, if \(m\) is even, then for every positive integer \(p\), the complete graph \(K_{2pn+1}\) can also be cyclically decomposed into the isomorphic copies of \(T\).
An aperiodic perfect map (APM) is an array with the property that each possible array of a given size, called a window, arises exactly once as a contiguous subarray in the array. In this paper, we give a construction method of an APM being a proper concatenation of some fragments of a given de Bruijn sequence. Firstly, we give a criterion to determine whether a designed sequence \(T\) with entries from the index set of a de Bruijn sequence can generate an APM. This implies a sufficient condition for being an APM. Secondly, two infinite families of APMs are given by constructions of corresponding sequences \(T\), respectively, satisfying the criterion.
We introduce a combinatorial shifting operation on multicomplexes that carries similar properties required for the ordinary shifting operation on simplicial complexes. A linearly colored simplicial complex is called shifted if its associated multicomplex is stable under defined operation. We show that the underlying simplicial subcomplex of a linearly shifted simplicial complex is shifted in the ordinary sense, while the ordinary and linear shiftings are not interrelated in general. Separately, we also prove that any linearly shifted complex must be shellable with respect to the order of its facets induced by the linear coloring. As an application, we provide a characterization of simple graphs whose independence complexes are linearly shifted. The class of graphs obtained constitutes a superclass of threshold graphs.
A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.
Explicit expressions for all the primitive idempotents in the ring \(R_{2^n} = {F}_q[x]/(x^{2^n} – 1)\), where \(q\) is an odd prime power, are obtained. Some lower bounds on the minimum distances of the irreducible cyclic codes of length \(2^n\) over \({F}_q\) are also obtained.
In this note we prove that all connected Cayley graphs of every finite group \(Q \times H\) are \(1\)-factorizable, where \(Q\) is any non-trivial group of \(2\)-power order and \(H\) is any group of odd order.
A graph \(G\) is called super vertex-magic total labelings if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(v) + \sum_{u \sim v} f(vu) = C\), where the sum is over all vertices \(u\) adjacent to \(v\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\), \(f(E(G)) = \{|V(G)|+1,|V(G)|+2,\ldots,|V(G)|+|E(G)|\}\). \({The Knödel graphs}\) \(W_{\Delta,n}\) have even \(n \geq 2\) vertices and degree \(\Delta\), \(1 \leq \Delta \leq \lfloor\log_2 n\rfloor\). The vertices of \(W_{\Delta,n}\) are the pairs \((i,j)\) with \(i = 1,2\) and \(0 \leq i \leq n/2-1\). For every \(j\), \(0 \leq j \leq n/2-1\), there is an edge between vertex \((1,j)\) and every vertex \((2,(j+2^k-1) \mod (n/2))\), for \(k=0,\ldots,\Delta-1\). In this paper, we show that \(W_{3,n}\) is super vertex-magic for \(n \equiv 0 \mod 4\).
Evolutionary graphs were initially proposed by Lieberman \(et \;al\). and evolutionary dynamics on two levels are recently introduced by Traulsen et al. We now introduce a new type of evolutionary dynamics,evolutionary graphs on two levels, and the fixation probability is analyzed. Some interesting results, evolutionary graphs on two levels are more stable than single level evolutionary graphs, are obtained in this paper.
A vertex \(k\)-ranking of a graph \(G\) is a function \(c: V(G) \to \{1,\ldots,k\}\) such that if \(c(u) = c(v)\), \(u,v \in V(G)\), then each path connecting vertices \(u\) and \(v\) contains a vertex \(w\) with \(c(w) > c(u)\). If each vertex \(v\) has a list of integers \(L(v)\) and for a vertex ranking \(c\) it holds \(c(v) \in L(v)\) for each \(v \in V(G)\), then \(c\) is called an \(L\)-list \(k\)-ranking, where \(\mathcal{L} = \{L(v) : v \in V(G)\}\). In this paper, we investigate both vertex and edge (vertex ranking of a line graph) list ranking problems. We prove that both problems are NP-complete for several classes of acyclic graphs, like full binary trees, trees with diameter at most \(4\), and comets. The problem of finding vertex (edge) \(\mathcal{L}\)-list ranking is polynomially solvable for paths and trees with a bounded number of non-leaves, which includes trees with diameter less than \(4\).
In this paper we determine unique graph with largest spectral radius among all tricyclic graphs with \(n\) vertices and \(k\) pendant edges.
A new proof is given to the following result of ours. Let \(G\) be an outerplanar graph with maximum degree \(\Delta \geq 3\). The chromatic number \(\chi(G^2)\) of the square of \(G\) is at most \(\Delta+2\), and \(\chi(G^2) = \Delta+1\) if \(\Delta \geq 7\).
Some designs using the action of the linear fractional groups \(L_2(q)\), \(q = 11, 13, 16, 17, 19, 23\) are constructed. We will show that \(L_2(q)\) or its automorphism group acts as the full automorphism group of each of the constructed designs except in the case \(q = 16\). For designs constructed from \(L_2(16)\), we will show that \(L_2(16)\), \(L_2(16) : 2\), \(L_2(16) : 4\) or \(S_{17}\) can arise as the full automorphism group of the design.
For odd \(n \geq 5\), the Flower Snark \(F_n = (V, E)\) is a simple undirected cubic graph with \(4n\) vertices, where \(V = \{a_i : 0 \leq i \leq n-1\} \cup \{b_i : 0 \leq i \leq n-1\} \cup \{c_i : 0 \leq i \leq 2n-1\}\) and \(E = \{b_ib_{(i+1)\mod(n)}: 0 \leq i \leq n-1\} \cup \{c_ic_{(i+1)\mod(2n)} : 0 \leq i \leq 2n-1\} \cup \{a_ib_i,a_ic_i,a_ic_{n+i} : 0 \leq i \leq n-1\}\). For \(n = 3\) or even \(n \geq 4\), \(F_n\) is called the related graph of Flower Snark. We show that the crossing number of \(F_n\) equals \(n – 2\) if \(3 \leq n \leq 5\), and \(n\) if \(n \geq 6\).
A subset \(S\) of the vertex set of a graph \(G\) is called acyclic if the subgraph it induces in \(G\) contains no cycles. We call \(S\) an acyclic dominating set if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by \(\gamma_a(G)\), is called the acyclic domination number of \(G\). A graph \(G\) is \({2-diameter-critical}\) if it has diameter \(2\) and the deletion of any edge increases its diameter. In this paper, we show that for any positive integers \(k\) and \(d \geq 3\), there is a \(2\)-diameter-critical graph \(G\) such that \(\delta(G) = d\) and \(\gamma_a(G) – \delta(G) \geq k\), and our result answers a question posed by Cheng et al. in negative.
A function \(f: V \to \{1,\ldots,k\}\) is a broadcast coloring of order \(k\) if \(\pi(u) = \pi(v)\) implies that the distance between \(u\) and \(v\) is more than \(\pi(u)\). The minimum order of a broadcast coloring is called the broadcast chromatic number of \(G\), and is denoted \(\chi_b(G)\). In this paper we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chromatic numbers. While there is a polynomial-time algorithm to determine whether \(\chi_b(G) \leq 3\), we show that it is \(NP\)-hard to determine if \(\chi_b(G) \leq 4\). We also determine the maximum broadcast chromatic number of a tree, and show that the broadcast chromatic number of the infinite grid is finite.
A connected graph \(G = (V, E)\) is said to be \((a,d)\)-antimagic if there exist positive integers \(a,d\) and a bijection \(f : E \to \{1,2,\ldots,|E|\}\) such that the induced mapping \(g_f : V \to \mathbb{N}\), defined by \(g_f(v) = \sum f(uv)\),\({uv \in E(G)}\) is injective and \(g_f(V) = \{a,a+d,\ldots,a+(|V|-1)d\}\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that the generalized Petersen graph \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 8\).
In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and \(4n\) subscripted Tribonacci sequences, \(\{T_{4n}\}\), and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally, we represent relationships between these sequences and permanents of certain matrices.
Let \(\mathcal{K} = (K_{ij})\) be an infinite lower triangular matrix of non-negative integers such that \(K_{i0} = 1\) and \(K_{ii} \geq 1\) for \(i \geq 0\). Define a sequence \(\{V_i(\mathcal{K})\}_{m\geq0}\) by the recurrence \(V_{i+1}(\mathcal{K}) = \sum_{j=0}^m K_{mj}V_j(\mathcal{K})\) with \(V_0(\mathcal{K}) = 1\). Let \(P(n;\mathcal{K})\) be the number of partitions of \(n\) of the form \(n = p_1 + p_2 + p_3 + p_4 + \cdots\) such that \(p_j \geq \sum_{i\geq j} K_{ij}p_{i+1}\) for \(j \geq 1\) and let \(P(n;V(\mathcal{K}))\) denote the number of partitions of \(n\) into summands in the set \(V(\mathcal{K}) = \{V_1(\mathcal{K}), V_2(\mathcal{K}), \ldots\}\). Based on the technique of MacMahon’s partitions analysis, we prove that \(P(n;\mathcal{K}) = P(n;V(\mathcal{K}))\) which generalizes a recent result of Sellers’. We also give several applications of this result to many classical sequences such as Bell numbers, Fibonacci numbers, Lucas numbers, and Pell numbers.
Given the number of vertices \( n \), labelled graphs can easily be generated uniformly at random by simply selecting each edge independently with probability \( \frac{1}{2} \). With \( \frac{n(n-1)}{2} \) processors, this takes constant parallel time. In contrast, the problem of uniformly generating unlabelled graphs of size \( n \) is not so straightforward. In this paper, we describe an efficient parallelisation of a classic algorithm of Dixon and Wilf for the uniform generation of unlabelled graphs on \( n \) vertices. The algorithm runs in \( O(\log n) \) expected time on a CREW PRAM using \( n^2 \) processors.
We discuss a parallel programming method for solving the maximum clique problem. We use the partitioned shared memory programming language, Unified Parallel \(C\), for the parallel implementation. The problem of load balancing is discussed and the steal stack is used to solve this problem. Implementation details are provided.
A \( 4 \)-cycle system of order \( n \) is said to be almost resolvable provided its \( 4 \)-cycles can be partitioned into \( \frac{n-1}{2} \) almost parallel classes (i.e., \( \frac{n-1}{4} \) vertex-disjoint \( 4 \)-cycles) and a half parallel class (i.e., \( \frac{n-1}{8} \) vertex-disjoint \( 4 \)-cycles). We construct an almost resolvable \( 4 \)-cycle system of every order \( n \equiv 1 \pmod{8} \) except \( 9 \) (for which no such system exists) and possibly \( 33, 41, \) and \( 57 \).
Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., \( (v, 2k, \lambda) \)-splitting BIBDs; we give the spectrum of \( (v, 2 \times 4, \lambda) \)-splitting BIBDs.
In this paper, we first present new proofs, much shorter and much simpler than can be found elsewhere, of two facts about Hypercubes: that for the \( d \)-dimensional Hypercube, there exist sets of paths by which any \({permutation\; routing}\) task may be accomplished in at most \( 2d – 1 \) steps without queueing; and, when \( d \) is even, there exists an edge decomposition of the Hypercube into precisely \( \frac{d}{2} \) edge-disjoint Hamiltonian cycles. The permutation routing paths are computed off-line. Whether or not these paths may be computed by an online parallel algorithm in \( O(d) \)-time has long been an open question. We conclude by speculating on whether the use of a Hamiltonian decomposition of the Hypercube might lead to such an algorithm.
The search for special substructures in combinatorial objects that have a lot of symmetry, such as searching for maximal partial ovoids or spreads in generalized quadrangles, can often be translated to a well-known algorithmic problem, such as a maximum clique problem in a graph. These problems are typically NP-hard. However, using standard backtracking strategies together with pruning techniques based on problem-specific properties, it is possible to obtain non-trivial results which are mathematically interesting. In some cases, heuristic techniques can also lead to interesting results. In this paper, we describe some techniques as well as new results obtained for maximal partial ovoids and spreads in generalized quadrangles.
Built on earlier works of Larcombe on a certain class of non-terminating expansions of the sine function, we set up two new \( {_{}{3}F_2} \) summation formulas via integration.
In this paper, we investigate exhaustively the cyclically indecomposable triple systems \( TS_\lambda(v) \) for \( \lambda = 2, v \leq 33 \) and \( \lambda = 3, v \leq 21 \), and we identify the decomposable ones. We also construct, by using Skolem-type and Rosa-type sequences, cyclically indecomposable two-fold triple systems \( TS_2(v) \) for all admissible orders. Further, we investigate exhaustively all cyclic \( TS_2(v) \) that are constructed by Skolem-type and Rosa-type sequences up to \( v \leq 45 \) for indecomposability.
We show that if the independence number of a graph is \( \alpha \), then the eternal security number of the graph is at most \( \binom{\alpha+1}{2} \), solving a problem stated by Goddard, Hedetniemi, and Hedetniemi \([JCMCC, \text{ vol. } 52, \text{ pp. } 160-180]\).
Let \( n \) be a natural number. We obtain convolution-type formulas for the total number of parts in all partitions of \( n \) of several different kinds.
In this paper, we establish a doubling method to construct inequivalent Hadamard matrices of order \( 2n \), from Hadamard matrices of order \( n \). Our doubling method uses heavily the symmetric group \( S_n \), where \( n \) is the order of a Hadamard matrix. We improve the efficiency of the method by introducing some group-theoretical heuristics. Using the doubling method in conjunction with the standard 4-row profile criterion, we have constructed several millions of new inequivalent Hadamard matrices of orders \(48, 56, 64, 72, 80, 88, 96,\) and several hundreds of inequivalent Hadamard matrices of orders 672 and 856. The Magma code segments, included in this paper, allow one to compute many more inequivalent Hadamard matrices of the above orders and all other orders of the form \( 8t \).
In this paper, we determine analytically the number of balanced, unlabelled, 3-member covers of an unlabelled finite set, which is then used to find the number of non-isomorphic optimal lottery sets of cardinality three. We also determine numerically the number of non-isomorphic optimal playing sets for lotteries in which a single correct number is required to win a prize.
A fire breaks out on a graph \( G \) and then \( f \) firefighters protect \( f \) vertices. At each subsequent interval, the fire spreads to all adjacent unprotected vertices, and firefighters protect \( f \) unburned vertices. Let \( f_G \) be the minimum number of firefighters needed to contain a fire on graph \( G \). If the triangular grid goes unprotected to time \( t = k \) when \( f_G \) firefighters arrive and begin protecting vertices, the fire can be contained by time \( t = 18k + 3 \) with at most \( 172k^2 + 58k + 5 \) vertices burned.
A construction is given for a Restricted Sarvate-Beam Triple System in the case \( v = 8 \). This is the extremal case, since a Restricted SB Triple System cannot exist for \( v > 8 \).
A \( t \)-\((v, k, \lambda) \) covering is a set of blocks of size \( k \) such that every \( t \)-subset of a set of \( v \) points is contained in at least \( \lambda \) blocks. The cardinality of the set of blocks is the size of the covering. The covering number \( C_\lambda(v, k, t) \) is the minimum size of a \( t \)-\((v, k, \lambda) \) covering. In this article, we find upper bounds on the size of \( t \)-\((v, k, 2) \) coverings for \( t = 3, 4 \), \( k = 5, 6 \), and \( v \leq 18 \). Twelve of these bounds are the exact covering numbers.
A tournament \(T = (V, A)\) is \({arc-traceable}\) if for each arc \(xy \in A\), \(xy\) lies on a directed path containing all the vertices of \(V\), i.e., a hamiltonian path. In this paper, we give two extremal results related to arc-traceability in tournaments. First, we show that a non-arc-traceable tournament \(T\) which is \(m\)-arc-strong must have at least \(2^{m+1}+4m-3\) vertices, and we construct an example that shows that this result is best possible. Next, we consider the maximum number of arcs in a strong tournament that are not part of any hamiltonian path. We use the structure of non-arc-traceable tournaments to prove that no strong tournament contains more than \(\frac{n^2-4n+3}{8}\) arcs that are not part of a hamiltonian path, and we give the unique example that shows that this bound is best possible.
Minimal blocking sets of class \([h,k]\) with respect to the external lines to an elliptic quadric of \(\text{PG}(3,q)\), \(q \geq 5\) prime, are characterized.
For every integer \(c\) and every positive integer \(k\), let \(n = r(c, k)\) be the least integer, provided that it exists, such that for every coloring
\[\Delta: \{1,2,\ldots,n\} \rightarrow \{0,1\},\]
there exist three integers, \(x_1, x_2, x_3\), (not necessarily distinct) such that
\[\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\]
and
\[x_1+x_2+c= kx_3.\]
If such an integer does not exist, then let \(r(c, k) = \infty\). The main result of this paper is that
\[r(c,2) =
\begin{cases}
|c|+1 & \text{if } c \text{ is even} \\
\infty & \text{if } c \text{ is odd}
\end{cases}\]
for every integer \(c\). In addition, a lower bound is found for \(r(c, k)\) for all integers \(c\) and positive integers \(k\) and linear upper and lower bounds are found for \(r(c, 3)\) for all positive integers \(c\).
Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\) with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0,3 \pmod 4\). The conjecture has been shown true for \(n = 3,5,6,7,4k\). In this paper, the conjecture is shown to be true for \(n = 9\).
In this paper, we define the hyperbolic modified Pell functions by the modified Pell sequence and classical hyperbolic functions. Afterwards, we investigate the properties of the modified Pell functions.
Deza and Grishukhin studied \(3\)-valent maps \(M_n{(p,q)}\) consisting of a ring of \(n\) \(g\)-gons whose inner and outer domains are filled by \(p\)-gons. They described the conditions for \(n, p, q\) under which such a map may exist and presented several infinite families of them. We extend their results by presenting several new maps concerning mainly large values of \(n\) and \(q\).
A simple, undirected \(2\)-connected graph \(G\) of order \(n\) belongs to the class \(\mathcal{B}(n,\theta)\), \(\theta \geq 0\) if \(2(d(x) + d(y) + d(z)) \geq 3(n – 1 – \theta)\) holds for all independent triples \(\{x,y,z\}\) of vertices. It is known (Bondy’s theorem for \(2\)-connected graphs) that \(G\) is hamiltonian if \(\theta = 0\). In this paper we give a full characterization of graphs \(G\) in \(\mathcal{B}(n,\theta)\), \(\theta \leq 2\) in terms of their dual hamiltonian closure.
Two classes of regular Cayley maps, balanced and antibalanced, have long been understood, see \([12,11]\). A recent generalization is that of an e-balanced map, see \([7,2,5,8]\). These maps can be described using the power function introduced in \([4]\); e-balanced maps are the ones with constant power functions on the generating set. In this paper we examine a further generalization to the situation where the power function alternates between two values.
In this paper, we obtain the spectral norm and eigenvalues of circulant matrices with Horadam’s numbers. Furthermore, we define the semicirculant matrix with these numbers and give the Euclidean norm of this matrix.
We denote by \(G(n)\) the graph obtained by removing a Hamilton cycle from the complete graph \(K_n\). In this paper, we calculate the lower bound for the minimum number of monochromatic triangles in any \(2\)-edge coloring of \(G(n)\) using the weight method. Also, by explicit constructions, we give an upper bound for the minimum number of monochromatic triangles in \(2\)-edge coloring of \(G(n)\) and the difference between our lower and upper bound is just two.
In this paper, it is proved that the \(h\)-chromatic uniqueness of the linear \(h\)-hypergraph consisting of two cycles of lengths \(p\) and \(q\) having \(r\) edges in common when \(p=q\), \(2 \leq r \leq p-2\), and \(h \geq 3\). We also obtain the chromatic polynomial of a connected unicyclic linear \(h\)-hypergraph and show that every \(h\)-uniform cycle of length three is not chromatically unique for \(h \geq 3\).
The projection of binary linear block codes of length \(4m\) on \(\mathbb{F}_4^m\) is considered. Three types of projections, namely projections \(SE\), \(E\), and \(O\) are introduced. The BCH codes, Golay codes, Reed-Muller codes, and the quadratic residue code \(q_{32}\) are examined.
The hyper Wiener index of a connected graph \(G\) is defined as
\(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2}\sum_{(u,v) \in V(G)} d(u,v)^2\) where \(d(u, v)\) is the distance between vertices \(u,v \in V(G)\).
In this paper we find an exact expression for hyper Wiener index of \(HC_6[p, q]\), the zigzag polyhex nanotori.
In this paper, we classify all optimal linear \([n, n/2]\) codes over \(\mathbb{Z}_4\) up to length \(n = 8\), and determine the number of optimal codes which are self-dual and formally self-dual. Optimal codes with linear binary images are identified. In particular, we show that for length \(8\), there are nine optimal codes for the Hamming distance, one optimal code for the Lee distance, and two optimal codes for the Euclidean distance.
In this paper, we show that if \(k \geq \frac{v+2}{4}\), where \(v\) denotes the order of a graph, a non-bipartite graph \(G\) is \(k\)-extendable if and only if it is \(2k\)-factor-critical. If \(k \geq \frac{v-3}{4}\), a graph \(G\) is \(k\)-extendable if and only if it is \((2k+1)\)-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron \([3]\) and Yu \([11]\).
The edge-neighbor-scattering number of a graph \(G\) is defined to be \(EN_S(G) = \max\limits_{S\subseteq E(G)}\{w(G/S) -\mid |S|\}\) where \(S\) is any edge-cut-strategy of \(G\), \(w(G/S)\) is the number of the components of \(G/S\). In this paper, we give edge-neighbor-scattering number of some special classes of graphs, and then mainly discuss the general properties of the parameter.
Let \(F(x,y) = ax^2 + bxy + cy^2\) be a binary quadratic form of discriminant \(\Delta = b^2 – 4ac\) for \(a,b,c \in \mathbb{Z}\), let \(p\) be a prime number and let \(\mathbb{F}_p\) be a finite field. In this paper we formulate the number of integer solutions of cubic congruence \(x^3 + ax^2 + bx + c \equiv 0 \pmod{p}\) over \(\mathbb{F}_p\), for two specific binary quadratic forms \(F_1^k(x,y) = x^2 + kxy + ky^2\) and \(F_2^k(x,y) = kx^2 + kxy + k^2y^2\) for integer \(k\) such that \(1 \leq k \leq 9\). Later we consider representation of primes by \(F_1^k\) and \(F_2^k\).
A subset \(S \subseteq V(G)\) is independent if no two vertices of \(S\) are adjacent in \(G\). In this paper we study the number of independent sets which meets the set of leaves in a tree. In particular we determine the smallest number and the largest number of these sets among \(n\)-vertex trees. In each case we characterize the extremal graphs.
A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\). Yasuhiro Fukuchi proved that the generalized Petersen graph \(P(n, 2)\) is super edge-magic for odd \(n \geq 3\). In this paper, we show that the generalized Petersen graph \(P(n,3)\) is super edge-magic for odd \(n \geq 5\).
For any integer \(k\), two tournaments \(T\) and \(T’\), on the same finite set \(V\) are \(k\)-similar, whenever they have the same score vector, and for every tournament \(H\) of size \(k\) the number of subtournaments of \(T\) (resp. \(T’\)) isomorphic to \(H\) is the same. We study the \(4\)-similarity. According to the decomposability, we construct three infinite classes of pairs of non-isomorphic \(4\)-similar tournaments.
In this paper, we define the Pell and Pell-Lucas \(p\)-numbers and derive the analytical formulas for these numbers. These formulas are similar to Binet’s formula for the classical Pell numbers.
A graph \(G\) is called resonant if the boundary of each face of \(G\) is an \(F\)-alternating closed trail with respect to some \(f\)-factor \(F\) of \(G\). We show that a plane bipartite graph \(G\) is resonant if and only if it is connected and each edge of \(G\) is contained in an \(f\)-factor and not in another \(f\)-factor.
Let \(P_k\) denote a path with \(k\) vertices and \(k-1\) edges. And let \(\lambda K_{n,n}\) denote the \(\lambda\)-fold complete bipartite graph with both parts of size \(n\). A \(P_k\)-decomposition \(\mathcal{D}\) of \(\lambda K_{n,n}\) is a family of subgraphs of \(\lambda K_{n,n}\) whose edge sets form a partition of the edge set of \(\lambda K_{n,n}\), such that each member of \(\mathcal{G}\) is isomorphic to \(P_k\). Necessary conditions for the existence of a \(P_k\)-decomposition of \(\lambda K_{n,n}\) are (i) \(\lambda n^2 \equiv 0 \pmod{k-1}\) and (ii) \(k \leq n+1\) if \(\lambda=1\) and \(n\) is odd, or \(k \leq 2n\) if \(\lambda \geq 2\) or \(n\) is even. In this paper, we show these necessary conditions are sufficient except for the possibility of the case that \(k=3\), \(n=15\), and \(k=28\).
We describe a technique for producing self-dual codes over rings and fields from symmetric designs. We give special attention to biplanes and determine the minimum weights of the codes formed from these designs. We give numerous examples of self-dual codes constructed including an optimal code of length \(22\) over \(\mathbb{Z}_4\) with respect to the Hamming metric from the biplane of order \(3\).
The distance graph \(G(S, D)\) has vertex set \(V(G(S, D)) = S \cup \mathbb{R}^n\) and two vertices \(u\) and \(v\) are adjacent if and only if their distance \(d(u, v)\) is an element of the distance set \(D \subseteq \mathbb{R}_+\).
We determine the chromatic index, the choice index, the total chromatic number and the total choice number of all distance graphs \(G(\mathbb{R}, D)\), \(G(\mathbb{Q}, D)\) and \(G(\mathbb{Z}, D)\) transferring a theorem of de Bruijn and Erdős on infinite graphs. Moreover, we prove that \(|D| + 1\) is an upper bound for the chromatic number and the choice number of \(G(S,D)\), \(S \subseteq \mathbb{R}\).
Some results on combinatorial aspects of block designs using the complementary property have been obtained. The results pertain to non-existence of partially balanced incomplete block (PBIB) designs and identification of new \(2\)-associate and \(3\)-associate PBIB designs. A method of construction of extended group divisible (EGD) designs with three factors using self-complementary rectangular designs has also been given. Some rectangular designs have also been obtained using self-complementary balanced incomplete block designs. Catalogues of EGD designs and rectangular designs obtainable from these methods of construction, with number of replications \(\leq 10\) and block size \(\leq 10\) have been prepared.
For any simple graph \(H\), let \(\sigma(H, n)\) be the minimum \(m\) so that for any realizable degree sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with sum of degrees at least \(m\), there exists an \(n\)-vertex graph \(G\) witnessing \(\pi\) that contains \(H\) as a weak subgraph. Let \(F_{k}\) denote the friendship graph on \(2k+1\) vertices, that is, the graph of \(k\) triangles intersecting in a single vertex. In this paper, for \(n\) sufficiently large, \(\sigma(F_{k},n)\) is determined precisely.
Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.
This paper studies families of self-orthogonal codes over \(\mathbb{Z}_4\). We show that the simplex codes (of Type \(a\) and Type \(\beta\)) are self-orthogonal. We answer the question of \(\mathbb{Z}_4\)-linearity for some codes obtained from projective planes of even order. A new family of self-orthogonal codes over \(\mathbb{Z}_4\) is constructed via projective planes of odd order. Properties such as self-orthogonality, weight distribution, etc. are studied. Finally, some self-orthogonal codes constructed from twistulant matrices are presented.
A complete paired comparison digraph \(D\) is a directed graph in which \(xy\) is an arc for all vertices \(x,y\) in \(D\), and to each arc we assign a real number \(0 \leq a \leq 1\) called a weight such that if \(xy\) has weight \(a\) then \(yx\) has weight \(1 – a\). We say that two vertices \(x, y\) dominate a third \(z\) if the weights on \(xz\) and \(yz\) sum to at least \(1\). If \(x\) and \(y\) dominate all other vertices in a complete paired comparison digraph, then we say they are a dominant pair. We construct the domination graph of a complete paired comparison digraph \(D\) on the same vertices as \(D\) with an edge between \(x\) and \(y\) if \(x\) and \(y\) form a dominant pair in \(D\). In this paper, we characterize connected domination graphs of complete paired comparison digraphs. We also characterize the domination graphs of complete paired comparison digraphs with no arc weight of \(.5\).
A graph \(G\) is a \((d,d+k)\)-graph, if the degree of each vertex of \(G\) is between \(d\) and \(d+k\). Let \(p > 0\) and \(d+k \geq 2\) be integers. If \(G\) is a \((d,d+k)\)-graph of order \(n\) with at most \(p\) odd components and without a matching \(M\) of size \(2|M| = n – p\), then we show in this paper that
Corresponding results for \(0 \leq p \leq 1\) and \(0 \leq k \leq 1\) were given by Wallis \([6]\), Zhao \([8]\), and Volkmann \([5]\).
Examples will show that the given bounds (i) and (ii) are best possible.
In this paper, we prove that the cycle \(C_n\) with parallel chords and the cycle \(C_n\) with parallel \(P_k\)-chords are cordial for any odd positive integer \(k \geq 3\) and for all \(n \geq 4\) except for \(n \equiv 4r + 2, r \geq 1\). Further, we show that every even-multiple subdivision of any graph \(G\) is cordial and we show that every graph is a subgraph of a cordial graph.
A hypergraph is linear if no two distinct edges intersect in more than one vertex. A long standing conjecture of Erdős, Faber, and Lovász states that if a linear hypergraph has \(n\) edges, each of size \(n\), then its vertices can be properly colored with \(n\) colors. We prove the correctness of the conjecture for a new, infinite class of linear hypergraphs.
We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.
Let \(G\) be a graph in which each vertex has been coloured using one of \(k\) colours, say \(c_1,c_2,\ldots,c_k\) If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices coloured \(c_i\), \(i = 1,2,\ldots,k\), and \(|n_i – n_j| \leq 1\) for any \(i,j \in \{1,2,\ldots,k\}\), then \(C\) is equitably \(k\)-coloured. An \(m\)-cycle decomposition \(C\) of a graph \(G\) is equitably \(k\)-colourable if the vertices of \(G\) can be coloured so that every \(m\)-cycle in \(C\) is equitably \(k\)-coloured. For \(m = 4,5\) and \(6\), we completely settle the existence problem for equitably \(2\)-colourable \(m\)-cycle decompositions of complete graphs and complete graphs with the edges of a \(1\)-factor removed.
Only the rotational tournament \(U_n\) for odd \(n \geq 5\), has the cycle \(C_n\) as its domination graph. To include an internal chord in \(C_n\), it is necessary for one or more arcs to be added to \(U_n\), in order to create the extended tournament \(U_n^+\). From this, the domination graph of \(U_t\), \(dom(U_n^+)\), may be constructed where \(C_k\), \(3 \leq k \leq n\), is a subgraph of \(dom(U_n^+)\). This paper explores the characteristics of the arcs added to \(U_n\) that are required to create an internal chord in \(C_n\).
We point out that restricted SB triple systems can only exist for \(v \leq 8\). The case \(v = 8\) is especially interesting since it is extremal in that the pair frequencies of the fifteen pairs not involving either \(1\) or \(2\) must be the frequencies \(2, 3, \dots, 16\), in some order.
Let \( a \) and \( b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:
\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even,} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd,}
\end{cases}
\]
\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even,} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd.}
\end{cases}
\]
For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short, \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+: V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and
\[
f^+(u) = \sum\{ f(u,v) : (u, v) \in E(G) \}.
\]
We investigate \( Q(a)P(b) \) super-edge-graceful labelings for three classes of \( (p,p+1) \)-graphs.
The Ramsey number \( R(C_p, C_q, C_r) \) is the smallest positive integer \( m \) such that no matter how one colors the edges of the \( K_m \) in red, white, and blue, there must be a red \( C_p \), a white \( C_q \), or a blue \( C_r \). In this work, we verified some known \( R(C_p, C_q, C_r) \) values and computed some new \( R(C_p, C_q, C_r) \) values. The results are based on computer algorithms.
A \( (p,q) \) graph \( G \) is total edge-magic if there exists a bijection \( f: V \cup E \to \{1, 2, \ldots, p+q\} \) such that for each \( e = (u,v) \in E \), we have \( f(u) + f(e) + f(v) \) as a constant. For a graph \( G \), denote \( M(G) \) the set of all total edge-magic labelings. The magic strength of \( G \) is the minimum of all constants among all labelings in \( M(G) \), denoted by \( \text{emt}(G) \). The maximum of all constants among \( M(G) \) is called the maximum magic strength of \( G \) and denoted by \( \text{eMt}(G) \).
Hegde and Shetty classify a magic graph as strong if \( \text{emt}(G) = \text{eMt}(G) \), ideal magic if \( 1 \leq \text{eMt}(G) – \text{emt}(G) \leq p \), and \(\textbf{weak magic}\) if \( \text{eMt}(G) – \text{emt}(G) > p \). A total edge-magic graph is called a super edge-magic if \( f(V(G)) = \{1, 2, \ldots, p\} \). The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.
For even codeword length \( n = 2k, k > 1 \) and alphabet size \( \sigma > 1 \), a family of comma-free codes is constructed with \({\left\lfloor \frac{\sigma^2}{3} \right\rfloor}^r \left( \sigma^2 – \left\lfloor \frac{\sigma^2}{3} \right\rfloor \right)^{k-r}\) codewords where \( 1 \leq r < k \). In particular, a new maximal comma-free code with \( n = 4 \) and \( \sigma = 4 \) is given by one of these codes.
If \( K \) is an \( r \)-clique of \( G \) and \( \chi(G) \) decreases by \( r \) upon the removal of all of the vertices in \( K \), then \( K \) is called a critical \( r \)-clique. Two critical cliques are completely independent provided that no vertex in one clique is adjacent to a vertex from the other. An infinite family of graphs is constructed which demonstrates that for every \( s, t \in \mathbb{N} \), there exists a vertex critical graph which admits a critical \( s \)-clique and a critical \( t \)-clique that are completely independent.
In this paper, we obtain a set of inequalities which are necessary conditions for the existence of balanced arrays of strength five, having \( m \) rows (constraints), and with two symbols. We discuss the use of these inequalities to obtain an upper bound on \( m \), and present some illustrative examples.
For a graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), let \( i(G) \) be the number of isolated vertices in \( G \). The \emph{isolated toughness} of \( G \) is defined as \(I(G) = \min\left\{\frac{|S|}{i(G-S)} \mid S \subseteq V(G), i(G-S) \geq 2 \right\},\)if \( G \) is not complete; and \( I(K_n) = n-1 \). In this paper, we investigate the existence of \([a, b]\)-factors in terms of this graph invariant. We proved that if \( G \) is a graph with \( \delta(G) \geq a \) and \( I(G) \geq a \), then \( G \) has a fractional \( a \)-factor. Moreover, if \( \delta(G) \geq a \), \( I(G) > (a-1) + \frac{a-1}{b} \), and \( G-S \) has no \( (a-1) \)-regular component for any subset \( S \) of \( V(G) \), then \( G \) has an \([a, b]\)-factor. The latter result is a generalization of Katerinis’ well-known theorem about \([a, b]\)-factors (P. Katerinis, Toughness of graphs and the existence of factors, \emph{Discrete Math}. 80(1990), 81-92).
We partition the set of spanning trees contained in the complete graph \( K_n \) into spanning trees contained in the complete bipartite graph \( K_{s,t} \). This classification shows that some properties of spanning trees in \( K_n \) can be derived from trees in \( K_{s,t} \). We use Abel’s binomial theorem and the formula for spanning trees in \( K_{s,t} \) to obtain a proof of Cayley’s theorem using partial derivatives. Some results concerning non-isomorphic spanning trees are presented. In particular, we count these trees for \( Q_3 \) and the Petersen graph.
We introduce the ring of ordinomials, which will be utilized in defining the partial chromatic ordinomials of infinite graphs with certain properties – a generalization of chromatic polynomials of finite graphs.
In algebraic contexts, Weyl group elements are usually represented in terms of generators and relations, where representation is not unique. For computational purposes, a more combinatorial representation for elements of classical Weyl groups as signed permutation vectors was introduced in [5]. This paper characterizes some special classes of Weyl group elements using this notation. These classes are especially useful for the study of symmetric spaces and their representations.
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A labeling \( f: V(G) \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E(G) \to \mathbb{Z}_2 \), defined by \( f^*(xy) = f(x) + f(y) \), for each edge \( xy \in E(G) \). For \( i \in \mathbb{Z}_2 \), let \(\text{v}_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}\) and \(\text{e}_f(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.\)A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert \text{v}_f(0) – \text{v}_f(1) \rvert \leq 1.\)The friendly index set of the graph \( G \), \( FI(G) \), is defined as \(\{ \lvert \text{e}_f(0) – \text{e}_f(1) \rvert : \text{the vertex labeling } f \text{ is friendly} \}.\)This is a generalization of graph cordiality. We introduce a graph construction called the root-union and investigate when gaps exist in the friendly index sets of root-unions of stars by cycles.
Proposed in 1942, the Graph Reconstruction Conjecture posits that every simple, finite, undirected graph with three or more vertices can be reconstructed up to isomorphism to the original graph, given the multiset of subgraphs produced by deleting each vertex along with its incident edges. Related to this Reconstruction Conjecture, existential reconstruction numbers, \( \exists rn(G) \), concern the minimum number of vertex-deleted subgraphs required to identify a graph up to isomorphism.We discuss the resulting data from calculating reconstruction numbers for all simple, undirected graphs with up to ten vertices. From this data, we establish the reasons behind all high existential reconstruction numbers (\( \exists rn(G) > 3 \)) for \( |V(G)| \leq 10 \) and identify new classes of graphs that have high reconstruction numbers for \( |V(G)| > 10 \).We also consider 2-reconstructibility—the ability to reconstruct a graph \( G \) from the multiset of subgraphs produced by deleting each combination of two vertices from \( G \). The 2-reconstructibility of all graphs with nine or fewer vertices was tested, identifying four graphs in this range with five vertices as the highest order of graphs that are not 2-reconstructible.
A weighing matrix \( W(n, k) \) of order \( n \) with weight \( k \) is an \( n \times n \) matrix with entries from \( \{0, 1, -1\} \) which satisfies \( WW^T = kI_n \). Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group \( G \) so that \( w_{g,h} = w_{gf,hf} \) for all \( g,h,f \) in \( G \). Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets.
We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a \( W(60, 25) \) developed over a non-solvable group; no solvable examples are known.
We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.
A \( (p,q) \)-graph \( G \) is said to be edge graceful if the edges can be labeled by \( 1, 2, \ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. In [7], we establish that every tree of odd order with one even vertex is edge-graceful. Mitchem and Simoson [19] introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for trees. We show that every tree of odd order with three even vertices is super edge-graceful.
It is known that there is not any non-trivial graph with vertices of distinct degrees, and any non-trivial graph must have at least two vertices of the same degree. In this article, we will consider the concept of \( P_3 \)-degree of vertices and will introduce a class of connected graphs with exactly two vertices of the same \( P_3 \)-degree. Also, the graphs with distinct \( P_3 \)-degree vertices will be constructed and it will be proven that for any \( n \geq 6 \) there is at least one graph of order \( n \), with distinct \( P_3 \)-degree vertices.
Let \( a, b \) be two positive integers. A \( (p, q) \)-graph \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful, or simply \( (a, b) \)-SEG, if there exist onto mappings \( f : E(G) \to Q(a) \) and \( f^* : V(G) \to P(b) \), where
\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even}, \\
\{0, \pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd},
\end{cases}
\]
\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even}, \\
\{0, \pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd},
\end{cases}
\]
such that \( f^*(v) = \sum_{uv \in E(G)} f(uv) \). We find the values of \( a \) and \( b \) for which the hypercube \( Q_n, n \leq 3 \), is \( (a, b) \)-SEG.
A map is a graph that admits an orientation of its edges so that each vertex has out-degree exactly \(1\). We characterize graphs which admit a decomposition into \(k\) edge-disjoint maps after: (1) the addition of any \(\ell\) edges; (2) the addition of some \(\ell\) edges. These graphs are identified with classes of \emph{sparse} graphs; the results are also given in matroidal terms.
A connected graph on three or more vertices is said to be irreducible if it has no leaves, and if each vertex has a unique neighbor set. A connected graph on one or two vertices is also said to be irreducible, and a disconnected graph is irreducible if each of its connected components is irreducible. In this paper, we study the class of irreducible graphs. In particular, we consider an algorithm that, for each connected graph \( \Gamma \), yields an irreducible subgraph \( I(\Gamma) \) of \( \Gamma \). We show that this subgraph is unique up to isomorphism. We also show that almost all graphs are irreducible. We then conclude by highlighting some structural similarities between \( I(\Gamma) \) and \( \Gamma \).
A Latin square of order \( n \) is an \( n \) by \( n \) array in which every row and column is a permutation of a set \( N \) of \( n \) elements. Let \( L = [l_{i,j}] \) and \( M = [m_{i,j}] \) be two Latin squares of even order \( n \), based on the same \( N \)-set. Define the superposition of \( L \) onto \( M \) to be the \( n \) by \( n \) array \( A = (l_{i,j}, m_{i,j}) \). When \( n \) is even, \( L \) and \( M \) are said to be \({nearly\; orthogonal}\) if the superposition of \( L \) onto \( M \) has every ordered pair \( (i, j) \) appearing exactly once except for \( i = j \), when the ordered pair appears \( 0 \) times and except for \( i – j = \frac{n}{2} \pmod{n} \), when the ordered pair appears \( 2 \) times. A set of \( t \) Latin squares of order \( 2m \) is called a set of \({mutually\; nearly\; orthogonal\; Latin \;squares}\) (MNOLS(\(2m\))) if the \( t \) Latin squares are pairwise nearly orthogonal. We provide two elementary proofs for results that were stated and proved earlier. We also provide some computer results and prove two recursive constructions for MNOLS. Using these results we show that there always exist \( 3 \) mutually nearly orthogonal Latin squares of order \( 2m \), for \( 2m \geq 358 \).
Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a dominating set if every vertex of \(V(G) – S\) is adjacent to some vertices in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). In this paper, we study the domination number of generalized Petersen graphs \(P(n,3)\) and prove that \(\gamma(P(n,3)) = n – 2\left\lfloor \frac{n}{4} \right\rfloor (n\neq 11)\).
The maximality of the Suzuki group \(\text{Sz}(K,a)\), \(K\) any commutative field of characteristic \(2\) admitting an automorphism \(\sigma\) such that \(x^{\sigma^2} = x^2\), in the symplectic group \(\text{PSp}_4(K)\), is proved.
Let \(k\) be a positive integer and \(G = (V, E)\) be a connected graph of order \(n\). A set \(D \subseteq V\) is called a \(k\)-dominating set of \(G\) if each \(x \in V(G) – D\) is within distance \(k\) from some vertex of \(D\). A connected \(k\)-dominating set is a \(k\)-dominating set that induces a connected subgraph of \(G\). The connected \(k\)-domination number of \(G\), denoted by \(\gamma_k^c(G)\), is the minimum cardinality of a connected \(k\)-dominating set. Let \(\delta\) and \(\Delta\) denote the minimum and the maximum degree of \(G\), respectively. This paper establishes that \(\gamma_k^c(G) \leq \max\{1, n – 2k – \Delta + 2\}\), and \(\gamma_k^c(G) \leq (1 + o_\delta(1))n \frac{ln[m(\delta+1)+2-t]}{m(\delta+1)+2-t}\), where \(m = \lceil \frac{k}{3} \rceil\), \(t = 3 \lceil \frac{k}{3} \rceil – k\), and \(o_\delta(1)\) denotes a function that tends to \(0\) as \(\delta \to \infty\). The later generalizes the result of Caro et al. in [Connected domination and spanning trees with many leaves. SIAM J. Discrete Math. 13 (2000), 202-211] for \(k = 1\).
A graph \(U\) is (induced)-universal for a class of graphs \({X}\) if every member of \({X}\) is contained in \(U\) as an induced subgraph. We study the problem of finding universal graphs with minimum number of vertices for various classes of bipartite graphs: exponential classes, bipartite chain graphs, bipartite permutation graphs, and general bipartite graphs. For exponential classes and general bipartite graphs we present a construction which is asymptotically optimal, while for the other classes our solutions are optimal in order.
The multicolor Ramsey number \(R_r(H)\) is defined to be the smallest integer \(n = n(r)\) with the property that any \(r\)-coloring of the edges of complete graph \(K_n\) must result in a monochromatic subgraph of \(K_n\) isomorphic to \(H\). In this paper, we study the case that \(H\) is a cycle of length \(2k\). If \(2k \geq r+1\) and \(r\) is a prime power, we show that \(R_r(C_{2k}) > {r^2+2k-r-1}\).
The bondage number \(b(D)\) of a digraph \(D\) is the cardinality of a smallest set of arcs whose removal from \(D\) results in a digraph with domination number greater than the domination number of \(D\). In this paper, we present some upper bounds on bondage number for oriented graphs including tournaments, and symmetric planar digraphs.
Let \(G\) be a connected graph. For a vertex \(v \in V(G)\) and an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the representation of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(r(v|\Pi) = (d(v, S_1), d(v, S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is said to be resolving if the \(k\)-vectors \(r(v|\Pi), v \in V(G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is called the partition dimension of \(G\), denoted by \(pd(G)\). A resolving \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\) is said to be connected if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is connected in \(G\). The minimum \(k\) for which there is a connected resolving \(k\)-partition of \(V(G)\) is called the connected partition dimension of \(G\), denoted by \(cpd(G)\). In this paper, the partition dimension as well as the connected partition dimension of the wheel \(W_n\) with \(n\) spokes are considered, by showing that \(\lceil (2n)^{1/3} \rceil \leq pd(W_n) \leq \lceil 2(n)^{1/2} \rceil +1\) and \(cpd(W_n) = \lceil (n+2)/3 \rceil\) for \(n \geq 4\).
Vertices \(x\) and \(y\) are called paired in tournament \(T\) if there exists a vertex \(z\) in the vertex set of \(T\) such that either \(x\) and \(y\) beat \(z\) or \(z\) beats \(x\) and \(y\). Vertices \(x\) and \(y\) are said to be distinguished in \(T\) if there exists a vertex \(z\) in \(T\) such that either \(x\) beats \(z\) and \(z\) beats \(y\), or \(y\) beats \(z\) and \(z\) beats \(x\). Two vertices are strictly paired (distinguished) in \(T\) if all vertices of \(T\) pair (distinguish) the two vertices in question. The \(p/d\)-graph of a tournament \(T\) is a graph which depicts strictly paired or strictly distinguished pairs of vertices in \(T\). \(P/d\)-graphs are useful in obtaining the characterization of such graphs as domination and domination-compliance graphs of tournaments. We shall see that \(p/d\)-graphs of tournaments have an interestingly limited structure as we characterize them in this paper. In so doing, we find a method of constructing a tournament with a given \(p/d\)-graph using adjacency matrices of tournaments.
Let \(G\) be a simple connected graph. The spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue of its adjacency matrix. In this paper, we obtain two lower bounds of \(\rho(G)\) by two different methods, one of which is better than another in some conditions.
In this note we compute the chromatic polynomial of the Jahangir graph \(J_{2p}\) and we prove that it is chromatically unique for \(p=3\).
In this paper, we compute the PI and Szeged indices of some important classes of benzenoid graphs, which some of them are related to nanostructures. Some open questions are also included.
The detour \(d(i, j)\) between vertices \(i\) and \(j\) of a graph is the number of edges of the longest path connecting these vertices. The matrix whose \((i, j)\)-entry is the detour between vertices \(i\) and \(j\) is called the detour matrix. The half sum \(D\) of detours between all pairs of vertices (in a connected graph) is the detour index, i.e.,
\[D = (\frac{1}{2}) \sum\limits_j\sum\limits_i d(i,j)\]
In this paper, we computed the detour index of \(TUC_4C_8(S)\) nanotube.
We construct several new group divisible designs with block size five and with \(2, 3\), or \(6\) groups.
A list \((2,1)\)-labeling \(\mathcal{L}\) of graph \(G\) is an assignment list \(L(v)\) to each vertex \(v\) of \(G\) such that \(G\) has a \((2,1)\)-labeling \(f\) satisfying \(f(v) \in L(v)\) for all \(v\) of graph \(G\). If \(|L(v)| = k + 1\) for all \(v\) of \(G\), we say that \(G\) has a \(k\)-list \((2,1)\)-labeling. The minimum \(k\) taken over all \(k\)-list \((2,1)\)-labelings of \(G\), denoted \(\lambda_l(G)\), is called the list label-number of \(G\). In this paper, we study the upper bound of \(\lambda(G)\) of some planar graphs. It is proved that \(\lambda_l(G) \leq \Delta(G) + 6\) if \(G\) is an outerplanar graph or \(A\)-graph; and \(\lambda_l(G) \leq \Delta(G) + 9\) if \(G\) is an \(HA\)-graph or Halin graph.
In this paper, we give a necessary and sufficient condition for a \(3\)-regular graph to be cordial.
This paper deals with the interconnections between finite weakly superincreasing distributions, the Fibonacci sequence, and Hessenberg matrices. A frequency distribution, to be called the Fibonacci distribution, is introduced that expresses the core of the connections among these three concepts. Using a Hessenberg representation of finite weakly superincreasing distributions, it is shown that, among all such \(n\)-string frequency distributions, the Fibonacci distribution achieves the maximum expected codeword length.
We present some applications of wall colouring to scheduling issues. In particular, we show that the chromatic number of walls has a very clear meaning when related to certain real-life situations.
Let \(G\) be a connected graph. For \(S \subseteq V(G)\), the geodetic closure \(I_G[S]\) of \(S\) is the set of all vertices on geodesics (shortest paths) between two vertices of \(S\). We select vertices of \(G\) sequentially as follows: Select a vertex \(v_1\) and let \(S_1 = \{v_1\}\). Select a vertex \(v_2 \neq v_1\) and let \(S_2 = \{v_1, v_2\}\). Then successively select vertex \(v_i \notin I_G[S_{i-1}]\) and let \(S_i = \{v_1, v_2, \ldots, v_i\}\). We define the closed geodetic number (resp. upper closed geodetic number) of \(G\), denoted \(cgn(G)\) (resp. \(ucgn(G)\)), to be the smallest (resp. largest) \(k\) whose selection of \(v_1, v_2, \ldots, v_k\) in the given manner yields \(I_G[S_k] = V(G)\). In this paper, we show that for every pair \(a, b\) of positive integers with \(2 \leq a \leq b\), there always exists a connected graph \(G\) such that \(cgn(G) = a\) and \(ucgn(G) = b\), and if \(a < b\), the minimum order of such graph \(G\) is \(b\). We characterize those connected graphs \(G\) with the property: If \(cgn(G) < k < ucgn(G) = 6\), then there is a selection of vertices \(v_1, v_2, \ldots, v_k\) as in the above manner such that \(I_G[S_k] = V(G)\). We also determine the closed and upper closed geodetic numbers of some special graphs and the joins of connected graphs.
Let \(G\) be a graph with \(n\) vertices and suppose that for each vertex \(v\) in \(G\), there exists a list of \(k\) colors, \(L(v)\), such that there is a unique proper coloring for \(G\) from this collection of lists, then \(G\) is called a uniquely \(k\)-list colorable graph. We say that a graph \(G\) has the property \(M(k)\) if and only if it is not uniquely \(k\)-list colorable. M. Ghebleh and E. S. Mahmoodian characterized uniquely \(3\)-list colorable complete multipartite graphs except for the graphs \(K_{1*4,5}\), \(K_{1*5,4}, K_{1*4,4}\), \(K_{2,3,4}\), and \(K_{2,2,r}\), \(4 \leq r \leq 8\). In this paper, we prove that the graphs \(K_{1*4,5}\), \(K_{1*5,4}\), \(K_{1*4,4}\), and \(K_{2,3,4}\) have the property \(M(3)\).
Let \(G\) be a simple graph and \(f: V(G) \mapsto \{1, 3, 5, \ldots\}\) an odd integer valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a \((1, f)\)-odd factor if \(d_F(v) \in \{1, 3, \ldots, f(v)\}\) for all \(v \in V(G)\), where \(d_F(v)\) is the degree of \(v\) in \(F\). For an odd integer \(k\), if \(f(v) = k\) for all \(v\), then a \((1, f)\)-odd factor is called a \([1, k]\)-odd factor. In this paper, the structure and properties of a graph with a unique \((1, f)\)-odd factor is investigated, and the maximum number of edges in a graph of the given order which has a unique \([1, k]\)-odd factor is determined.
Erdős and Soifer \([3]\) and later Campbell and Staton \([1]\) considered a problem which was a favorite of Erdős \([2]\): Let \(S\) be a unit square. Inscribe \(n\) squares with no common interior point. Denote by \(\{e_1, e_2, \ldots, e_n\}\) the side lengths of these squares. Put \(f(n) = \max \sum\limits_{i=1}^n e_i\). And they discussed the bounds for \(f(n)\). In this paper, we consider its dual problem – covering a unit square with squares.
The well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph \(G\) in terms of the deficiency \(\max_{X \subseteq V(G)} \{ \omega_0(G – X) – |X| \}\) of \(G\), where \(\omega_0(H)\) denotes the number of odd components of \(H\). Let \(G’\) be the graph formed from \(G\) by subdividing (possibly repeatedly) a number of its edges. In this note we study the effect such subdivisions have on the difference between the size of a maximum matching in \(G\) and the size of a maximum matching in \(G’\).
In this paper, we give some necessary conditions for a prime graph. We also present some new families of prime graphs such as \(K_n \odot K_1\) is prime if and only if \(n \leq 7\), \(K_n \odot \overline{K_2}\) is prime if and only if \(n \leq 16\), and \(K_{m}\bigcup S_n\) is prime if and only if \(\pi(m+n-1) \geq m\). We also show that a prime graph of order greater than or equal to \(20\) has a nonprime complement.
Consider a lottery scheme consisting of randomly selecting a winning \(t\)-set from a universal \(m\)-set, while a player participates in the scheme by purchasing a playing set of any number of \(n\)-sets from the universal set prior to the draw, and is awarded a prize if \(k\) or more elements of the winning \(t\)-set occur in at least one of the player’s \(n\)-sets (\(1 \leq k \leq \{n,t\} \leq m\)). This is called a \(k\)-prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a \(k\)-prize, no matter which winning \(t\)-set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by \(L(m,n,t;k)\), and the number of such non-isomorphic sets is called the lottery characterisation number, denoted by \(\eta(m,n,t;k)\). In this paper, an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges \(2 \leq k \leq 4\), \(k \leq t \leq 11\), \(k \leq n \leq 12\), and \(\max\{n,t\} \leq m \leq 20\). In the process, \(32\) new lottery numbers are found, and bounds on a further \(31\) lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.
Beck’s coloring is studied for meet-semilattices with \(0\). It is shown that for such semilattices, the chromatic number equals the clique number.
The main result of this paper is an upper bound on the number of independent sets in a tree in terms of the order and diameter of the tree. This new upper bound is a refinement of the bound given by Prodinger and Tichy [Fibonacci Q., \(20 (1982), no. 1, 16-21]\). Finally, we give a sufficient condition for the new upper bound to be better than the upper bound given by Brigham, Chandrasekharan and Dutton [Fibonacci Q., \(31 (1993), no. 2, 98-104]\).
In this paper, it is shown that every extended directed triple system of order \(v\) can be embedded in an extended directed triple system of order \(n\) for all \(n \geq 2v\). This produces a generalization of the Doyen- Wilson theorem for extended directed triple systems.
A semigraph \(G\) is an ordered pair \((V,X)\) where \(V\) is a non-empty set whose elements are called vertices of \(G\) and \(X\) is a set of \(n\)-tuples (\(n > 2\)), called edges of \(G\), of distinct vertices satisfying the following conditions:
i) any edge \((v_1, v_2, \ldots, v_n)\) of \(G\) is the same as its reverse \((v_n, v_{n-1}, \ldots, v_1)\),and
ii) any two edges have at most one vertex in common.
Two edges are adjacent if they have a common vertex. \(G\) is edge complete if any two edges in \(G\) are adjacent. In this paper, we enumerate the non-isomorphic edge complete \((p,2)\)semigraphs.
Let \(G = (V, E)\) be a graph. A subset \(D \subseteq V\) is called a dominating set for \(G\) if for every \(v \in V – D\), \(v\) is adjacent to some vertex in \(D\). The domination number \(\gamma(G)\) is equal to \(\min \{|D|: D \text{ is a dominating set of } G\}\).
In this paper, we calculate the domination numbers \(\gamma(C_m \times C_n)\) of the product of two cycles \(C_m\) and \(C_n\) of lengths \(m\) and \(n\) for \(m = 5\) and \(n = 3 \mod 5\), also for \(m = 6, 7\) and arbitrary \(n\).
In this paper, we consider a certain second order linear recurrence and then give generating matriees for the sums of positively and negatively subscripted terms of this recurrence. Further, we use matrix methods and derive explicit. formulas for these sums.
For a simple and finite graph \(G = (V,E)\), let \(w_{\max}(G)\) be the maximum total weight \(w(E) = \sum_{e\in E} w(e)\) of \(G\) over all weight functions \(w: E \to \{-1,1\}\) such that \(G\) has no positive cut, i.e., all cuts \(C\) satisfy \(w(C) \leq 0\).
For \(r \geq 1\), we prove that \(w_{\max}(G) \leq -\frac{|V|}{2}\) if \(G\) is \((2r-1)\)-regular and \(w_{\max}(G) \leq -\frac{r|V|}{2r+1}\) if \(G\) is \(2r\)-regular. We conjecture the existence of a constant \(c\) such that \(w_{\max}(G) \leq -\frac{5|V|}{6} + c\) if \(G\) is a connected cubic graph and prove a special case of this conjecture. Furthermore, as a weakened version of this conjecture, we prove that \(w_{\max}(G) \leq -\frac{2|V|}{3}+\frac{2}{3}\) if \(G\) is a connected cubic graph.
Let \(G_i\) be the subgraph of \(G\) whose edges are in the \(i\)-th color in an \(r\)-coloring of the edges of \(G\). If there exists an \(r\)-coloring of the edges of \(G\) such that \(H_i \nsubseteq G_i\) for all \(1 \leq i \leq r\), then \(G\) is said to be \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). The multicolor Ramsey number \(R(H_1, H_2, \ldots, H_r)\) is the smallest integer \(n\) such that \(K_n\) is not \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). It is well known that \(R(C_m, C_4, C_4) = m + 2\) for sufficiently large \(m\). In this paper, we determine the values of \(R(C_m, C_4, C_4)\) for \(m \geq 5\), which show that \(R(C_m, C_4, C_4) = m + 2\) for \(m \geq 11\).
The proof of gracefulness for the Generalised Petersen Graph \(P_{8t,3}\) for every \(t \geq 1\), written by the same author (Graceful labellings for an infinite class of generalised Petersen graphs, Ars Combinatoria \(81 (2006)\), pp. \(247-255)\), requires the change of just one label, for the only case \(t = 5\).
For words of length \(n\), generated by independent geometric random variables, we study the average initial and end heights of the last descent in the word. In addition, we compute the average initial and end height of the last descent in a random permutation of \(n\) letters.
We construct a record-breaking binary code of length \(17\), minimal distance \(6\), constant weight \(6\), and containing \(113\) codewords.
The purpose of this note is to give the power formula of the generalized Lah matrix and show \(\mathcal{L}[x,y] = \mathcal{FQ}[x,y]\), where \(\mathcal{F}\) is the Fibonacci matrix and \(\mathcal{Q}[x,y]\) is the lower triangular matrix. From it, several combinatorial identities involving the Fibonacci numbers are obtained.
A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that some classes of separable graphs with a unique endvertex are set reconstructible and show that all graphs are set reconstructible if all \(2\)-connected graphs are set reconstructible.
We prove the following extension of the Erdős-Ginzburg-Ziv Theorem. Let \(m\) be a positive integer. For every sequence \(\{a_i\}_{i\in I}\) of elements from the cyclic group \(\mathbb{Z}_m\), where \(|I| = 4m – 5\) (where \(|I| = 4m – 3\)), there exist two subsets \(A, B \subseteq I\) such that \(|A \cap B| = 2\) (such that \(|A \cap B| = 1\)), \(|A| = |B| = m\), and \(\sum\limits_{i\in b} a_i = \sum\limits_{i\in b} b_i = 0\).
A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity \(\lambda’\) of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each component has at least two vertices. It has been shown by A. H. Esfahanian and S. L. Hakimi (On computing a conditional edge-connectivity of a graph. Information Processing Letters, 27(1988), 195-199] that \(\lambda'(G) \leq \xi(G)\) for any graph of order at least four that is not a star, where \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \text{ is an edge in } G\}\). A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \xi(G)\). This paper proves that the de Bruijn undirected graph \(UB(d,n)\) is \(\lambda’\)-optimal except \(UB(2,1)\), \(UB(3,1)\), and \(UB(2,3)\), and hence, is super edge-connected for \(n\geq 1\) and \(d\geq 2\).
The problem of graceful labeling of a particular class of trees called quasistars is considered. Such a quasistar is a tree \(Q\) with \(k\) distinct paths with lengths \(1, d+1, 2d+1, \ldots, (k-1)d+1\) joined at a unique vertex \(\theta\).
Thus, \(Q\) has \(1 + [1 + (d+1) + (2d+1) + \ldots + (k-1)d+1)] = 1+k +\frac{k(k-1)d}{2}\) vertices. The \(k\) paths of \(Q\) have lengths in arithmetic progression with common difference \(d\). It is shown that \(Q\) has a graceful labeling for all \(k \leq 6\) and all values of \(d\).
The average distance \(\mu(D)\) of a strong digraph \(D\) is the average of the distances between all ordered pairs of distinct vertices of \(D\). Plesnik \([3]\) proved that if \(D\) is a strong tournament of order \(n\), then \(\mu(D) \leq \frac{n+4}{6} + \frac{1}{n}\). In this paper we show that, asymptotically, the same inequality holds for strong bipartite tournaments. We also give an improved upper bound on the average distance of a \(k\)-connected bipartite tournament.
To measure the efficiency of a routing in network, Chung et al. [The forwarding index of communication networks. IEEE Trans. Inform. Theory, 33 (2) (1987), 224-232] proposed the notion of forwarding index and established an upper bound \((n – 1)(n – 2)\) of this parameter for a connected graph of order \(n\). This note improves this bound as \((n – 1)(n – 2) – (2n – 2 – \Delta\lfloor1+\frac{n-1}{\Delta}\rfloor)\) \(\lfloor \frac{n-1}{\Delta}\rfloor\) , where \(\Delta\) is the maximum degree of the graph \(G\). This bound is best possible in the sense that there is a graph \(G\) attaining it.
We study the spectral radius of unicyclic graphs with \(n\) vertices and edge independence number \(q\). In this paper, we show that of all unicyclic graphs with \(n\) vertices and edge independence number \(q\), the maximal spectral radius is obtained uniquely at \(\Delta_n(q)\), where \(\Delta_n(q)\) is a graph on \(n\) vertices obtained from the cycle \(C_3\) by attaching \(n – 2q + 1\) pendant edges and \(q – 2\) paths of length \(2\) at one vertex.
Let \(q\) be an odd prime power and \(p\) be an odd prime with \(gcd(p, g) = 1\). Let the order of \(g\) modulo \(p\) be \(f\) and \(gcd(\frac{p-1}{f}, q) = 1\). Here explicit expressions for all the primitive idempotents in the ring \(R_{2p^n} = GF(q)[x]/(x^{2p^n} – 1)\), for any positive integer \(n\), are obtained in terms of cyclotomic numbers, provided \(p\) does not divide \(\frac{q^f-1}{2p}\), if \(n \geq 2\). Some lower bounds on the minimum distances of irreducible cyclic codes of length \(2p^n\) over \(GF(q)\) are also obtained.
Let \(G\) be a connected multigraph with an even number of edges and suppose that the degree of each vertex of \(G\) is even. Let \((uv, G)\) denote the multiplicity of edge \((u,v)\) in \(G\). It is well known that we can obtain a halving of \(G\) into two halves \(G_1\) and \(G_2\), i.e. that \(G\) can be decomposed into multigraphs \(G_1\) and \(G_2\), where for each vertex \(v\), \(\deg(v, G_1) = \deg(v, G_2) = \frac{1}{2}\deg(v,G)\). It is also easy to see that if the edges with odd multiplicity in \(G\) induce no components with an odd number of edges, then we can obtain such a halving of \(G\) into two halves \(G_1\) and \(G_2\) that is well-spread, i.e. for each edge \((u,v)\) of \(G\), \(|\mu(uv, G_1) – \mu(uv, G_2)| \leq 1\). We show that if \(G\) is a \(\Delta\)-regular multigraph with an even number of vertices and with \(\Delta\) being even, then even if the edges with odd multiplicity in \(G\) induce components with an odd number of edges, we can still obtain a well-spread halving of \(G\) provided that we allow the addition/removal of a Hamilton cycle to/from \(G\). We give an application of this result to obtaining sports schedules such that multiple encounters between teams are well-spread throughout the season.
A fractional edge coloring of graph \(G\) is an assignment of a nonnegative weight \(w_M\) to each matching \(M\) of \(G\) such that for each edge \(e\) we have \(\sum_{M\ni e} w_M \geq 1\). The fractional edge coloring chromatic number of a graph \(G\), denoted by \(\chi’_f(G)\), is the minimum value of \(\sum_{M} w_M\) (where the minimum is over all fractional edge colorings \(w\)). It is known that for any simple graph \(G\) with maximum degree \(\Delta\), \(\Delta < \chi'_f(G) \leq \Delta+1\). And \(\chi'_f(G) = \Delta+1\) if and only if \(G\) is \(K_{2n+1}\). In this paper, we give some sufficient conditions for a graph \(G\) to have \(\chi'_f(G) = \Delta\). Furthermore, we show that the results in this paper are the best possible.
A subset \(D\) of the vertex set \(V\) of a graph is called an open odd dominating set if each vertex in \(V\) is adjacent to an odd number of vertices in \(D\) (adjacency is irreflexive). In this paper we solve the existence and enumeration problems for odd open dominating sets (and analogously defined even open dominating sets) in the \(m \times n\) grid graph and prove some structural results for those that do exist. We use a combination of combinatorial and linear algebraic methods, with particular reliance on the sequence of Fibonacci polynomials over \({GF}(2)\).
By introducing \(4\) colour classes in projective planes with non-Fano quads, discussion of the planes of small order is simplified.
Let \(G = (V, E)\) be a \(k\)-connected graph. For \(t \geq 3\), a subset \(T \subset V\) is a \((t,k)\)-shredder if \(|T| = k\) and \(G – T\) has at least \(t\) connected components. It is known that the number of \((t,k)\)-shredders in a \(k\)-connected graph on \(n\) nodes is less than \(\frac{2n}{2t – 3}\). We show a slightly better bound for the case \(k \leq 2t – 3\).
Let \(L\) and \(R\) be two graphs. For any positive integer \(n\), the Ehrenfeucht-Fraissé game \(G_n(L, R)\) is played as follows: on the \(i\)-th move, with \(1 \leq i \leq n\), the first player chooses a vertex on either \(L\) or \(R\), and the second player responds by choosing a vertex on the other graph. Let \(l_i\) be the vertex of \(L\) chosen on the \(i^{th}\) move, and let \(r_i\) be the vertex of \(R\) chosen on the \(i^{th}\) move. The second player wins the game iff the induced subgraphs \(L\{l_1,l_2,…,l_n\}\) and \(R\{r_1,r_2,…,r_n\}\) are isomorphic under the mapping sending \(l_i\) to \(r_i\). It is known that the second player has a winning strategy if and only if the two graphs, viewed as first-order logical structures (with a binary predicate E), are indistinguishable (in the corresponding first-order theory) by sentences of quantifier depth at most \(n\). In this paper we will give the first complete description of when the second player has a winning strategy for \(L\) and \(R\) being both paths or both cycles. The results significantly improve previous partial results.
By applying the method of generating function, the purpose of this paper is to give several summations of reciprocals related to \(l-th\) power of generalized Fibonacci sequences. As applications, some identities involving Fibonacci, Lucas numbers are obtained.
Bricks are polyominoes with labelled cells. The problem whether a given set of bricks is a code is undecidable in general. We consider sets consisting of square bricks only. We have shown that in this setting, the codicity of small sets (two bricks) is decidable, but \(15\) bricks are enough to make the problem undecidable. Thus the step from words to even simple shapes changes the algorithmic properties significantly (codicity is easily decidable for words). In the present paper we are interested whether this is reflected by quantitative properties of words and bricks. We use their combinatorial properties to show that the proportion of codes among all sets is asymptotically equal to \(1\) in both cases.
Let \(G_{n,m} = C_n \times P_m\), be the cartesian product of an \(n\)-cycle \(C_n\) and a path \(P_m\) of length \(m-1\). We prove that \(\chi'(G_{n,m}) = \chi'(G_{n,m}) = 4\) if \(m \geq 3\), which implies that the list-edge-coloring conjecture (LLECC) holds for all graphs \(G_{n,m}\).
Various authors have defined statistics on Dyck paths that lead to generalizations of the Catalan numbers. Three such statistics are area, maj, and bounce. Haglund, whe introduced the bounce statistic, gave an algebraic proof that \(n(n – 1)/2+\) area — bounce and maj have the same distribution on Dyck paths of order \(n\). We give an explicit bijective proof of the same result.
We develop a new type of a vertex labeling of graphs, namely \(2n\)-cyclic blended labeling, which is a generalization of some previously known labelings. We prove that a graph with this labeling factorizes the complete graph on \(2nk\) vertices, where \(k\) is odd and \(n, k > 1\).
Let \(D = (V, E)\) be a primitive digraph. The exponent of \(D\) at a vertex \(u \in V\), denoted by \(\text{exp}_D(u)\), is defined to be the least integer \(k\) such that there is a walk of length \(k\) from \(u\) to \(v\) for each \(v \in V\). Let \(V = \{v_1,v_2,\ldots ,v_n\}\). The vertices of \(V\) can be ordered so that \(\text{exp}_D(v_{i_1}) \leq \text{exp}_D(v_{i_2}) \leq \ldots \leq \text{exp}_D(v_{i_n})\). The number \(\text{exp}_D(v_{i_k})\) is called \(k\)-exponent of \(D\), denoted by \(\text{exp}_D(k)\). In this paper, we completely characterize \(1\)-exponent set of primitive, minimally strong digraphs with \(n\) vertices.
In \([4]\) H. Galana-Sanchez introduced the concept of kernels by monochromatic paths which generalize the concept of kernels. In \([6]\) they proved the necessary and sufficient conditions for the existence of kernels by monochromatic paths of the duplication of a subset of vertices of a digraph, where a digraph is without monochromatic directed circuits. In this paper we study independent by monochromatic paths sets and kernels by monochromatic paths of the duplication. We generalize result from \([6]\) for an arbitrary edge coloured digraph.
Let \(D = (V, E)\) be a primitive, minimally strong digraph. In \(1982\), J. A. Ross studied the exponent of \(D\) and obtained that \(\exp(D) \leq n + s(n – 8)\), where \(s\) is the length of a shortest circuit in \(D\) \([D]\). In this paper, the \(k\)-exponent of \(D\) is studied. Our principle result is that
\[
\exp_D(k) \leq \begin{cases}
k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\
k + s(n-3), & \text{if } s+1 \leq k \leq n,\\
\end{cases} \\.
\]
with equality if and only if \(D\) isomorphic to the diagraph \(D_{s,n}\) with vertex set \(V(D_{s,n})=\{v_1,v_2,\ldots,v_n\}\) and arc set \(E(D_{s,n})=\{(v_i,v_{i+1}):1\leq i\leq n-1\}\cap \{(v_s,v_1),(v_n,v_2)\}\). If \((s,n-1)\neq 1\),then
\[
\exp_D(k)< \begin{cases}
k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\
k + s(n-3), & \text{if } s+1 \leq k \leq n,
\end{cases} \\
\]
and if \((s,n-1)=1\), then \(D_{s, n}\) is a primitive, minimally strong digraph on \(n\) vertices with the \(k\)-exponent
\[
\exp_D(k)= \begin{cases}
k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\
k + s(n-3), & \text{if } s+1 \leq k \leq n,
\end{cases} \\
\]
Moreover, we provide a new proof of Theorem \(1\) in \([6]\) and Theorem \(2\) in \([14]\) by applying this result.
Given a finite projective plane of order \(n\). A quadrangle consists of four points \(A, B, C, D\), no three collinear. If the diagonal points are non-collinear, the quadrangle is called a non-Fano quad. A general sum of squares theorem is proved for the distribution of points in a plane containing a non-Fano quad, whenever \(n \geq 7\). The theorem implies that the number of possible distributions of points in a plane of order \(n\) is bounded for all \(n \geq 7\). This is used to give a simple combinatorial proof of the uniqueness of \(PP(7)\).
Let \(G = (V, E)\) be a graph with \(n\) vertices. The clique graph of \(G\) is the intersection graph \(K(G)\) of the set of all (maximal) cliques of \(G\) and \(K\) is called the clique operator. The iterated clique graphs \(K^*(G)\) are recursively defined by \(K^0(G) = G\) and \(K^i(G) = K(K^{i-1}(G))\), \(i > 0\). A graph is \(K\)-divergent if the sequence \(|V(K^i(G))|\) of all vertex numbers of its iterated clique graphs is unbounded, otherwise it is \(K\)-convergent. The long-run behaviour of \(G\), when we repeatedly apply the clique operator, is called the \(K\)-behaviour of \(G\).
In this paper, we characterize the \(K\)-behaviour of the class of graphs called \(p\)-trees, that has been extensively studied by Babel. Among many other properties, a \(p\)-tree contains exactly \(n – 3\) induced \(4\)-cycles. In this way, we extend some previous results about the \(K\)-behaviour of cographs, i.e., graphs with no induced \(P_4\)s. This characterization leads to a polynomial-time algorithm for deciding the \(K\)-convergence or \(K\)-divergence of any graph in the class.
In this paper, we obtain a general enumerating functional equation about rooted pan-fan maps on nonorientable surfaces. Based on this equation, an explicit expression of rooted pan-fan maps on the Klein bottle is given. Meanwhile, some simple explicit expressions with up to two parameters: the valency of the root face and the size for rooted one-vertexed maps on surfaces (Klein bottle, Torus, \(N_3\)) are provided.
Let us denote by \({EX}(m,n; \{C_4,\ldots,C_{2t}\})\) the family of bipartite graphs \(G\) with \(m\) and \(n\) vertices in its classes that contain no cycles of length less than or equal to \(2t\) and have maximum size. In this paper, the following question is proposed: does always such an extremal graph \(G\) contain a \((2t + 2)\)-cycle? The answer is shown to be affirmative for \(t = 2,3\) or whenever \(m\) and \(n\) are large enough in comparison with \(t\). The latter asymptotical result needs two preliminary theorems. First, we prove that the diameter of an extremal bipartite graph is at most \(2t\), and afterwards we show that its girth is equal to \(2t + 2\) when the minimum degree is at least \(2\) and the maximum degree is at least \(t + 1\).
Given a connected graph \( G \) with \( n \) vertices, a routing \( R \) is a collection of \( n(n-1) \) paths, one path \( R(x,y) \) for each ordered pair \( x, y \) of vertices. A routing is said to be vertex/edge-antisymmetric if for every pair \( x, y \) of vertices, the paths \( R(x,y) \) and \( R(y,x) \) are internally vertex/edge-disjoint. Compared to other types of routing found in the literature, antisymmetric routing is interesting from a practical point of view because it ensures greater network reliability.
For a given graph \( G \) and routing \( R \), the vertex/edge load of \( G \) with respect to \( R \) is the maximum number of paths passing through any vertex/edge of \( G \). The \({vertex/edge-forwarding-index}\) of a graph is the minimum vertex/edge load taken over all routings. If routing \( R \) is vertex/edge-antisymmetric, we talk about \({antisymmetric-indices}\).
Several results exist in the literature for the forwarding-indices of graphs. In this paper, we derive upper and lower bounds for the antisymmetric-indices of graphs in terms of their connectivity or minimum degree. These bounds are often the best possible. Whenever this is the case, a network that meets the corresponding bound is described. Several related conjectures are proposed throughout the paper.
A tree \( R \) such that after deleting all leaves we obtain a path \( P \) is called a \({caterpillar}\). The path \( P \) is called the \({spine}\) of the caterpillar \( R \). If the spine has length 3 and \( R \) on \( 2n \) vertices contains vertices of degrees \( r \), \( s \), \( t \), \( 2 \), where \( 2 < r, s, t < n \), then we say that \( R \) is an \( (r, s, t, 2) \)-\({caterpillar}\) of diameter 5. We completely characterize \( (r, s, t, 2) \)-caterpillars of diameter 5 on \( 4k+2 \) vertices that factorize the complete graph \( K_{4k+2} \).
This paper studies convex geometric graphs in which no path of length 3 self-intersects. A main result gives a decomposition of such graphs into induced outerplanar graph drawings. The resulting structure theorem is then used to compute a sharp, linear upper bound on the size of the edge set in terms of the number of vertices and the number and type of graphs in the decomposition. The paper also shows that though locally outerplanar graphs have hereditary properties, no graph property that is closed under the taking of minors can hold for all locally outerplanar graphs. Each of these results is generalized to convex geometric graphs in which no path of length \( k \) self-intersects.
By means of the partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apéry numbers will be established, including the binomial-harmonic number identity associated with Beukers’ conjecture on Apéry numbers.
Previous work on a certain class of non-terminating expansions of the sine function leads directly to a new result for associated infinite series by straightforward integration. A general identity is established, particular cases verified and two proofs of its hypergeometric form given.
An oriented graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let \( H \) be a 2-stratified oriented graph rooted at some blue vertex. An \( H \)-coloring of an oriented graph \( D \) is a red-blue coloring of the vertices of \( D \) in which every blue vertex \( v \) belongs to a copy of \( H \) rooted at \( v \) in \( D \). The \( H \)-domination number \( \gamma_H(D) \) is the minimum number of red vertices in an \( H \)-coloring of \( D \). We investigate \( H \)-colorings in oriented graphs where \( H \) is the red-red-blue directed path of order 3. Relationships between the \( H \)-domination number \( \gamma_H \) and both the domination number \( \gamma \) and open domination \( \gamma_o \), in oriented graphs are studied. It is shown that \( \gamma(D) \leq \gamma_H(D) \leq \gamma_o(D) \leq \lfloor \frac{3\gamma_H(D)}{2} \rfloor \) for every oriented graph \( D \). All pairs of positive integers that can be realized as (1) domination number and \( H \)-domination number and (2) the \( H \)-domination number and open domination number of some oriented graph are determined. Sharp bounds are established for the \( H \)-domination number of an \( r \)-regular oriented graph in terms of \( r \) and its order.
Defining sets of balanced incomplete block designs (BIBDs) were introduced by Ken Gray. Various authors have since identified minimal defining sets of particular BIBDs or classes of BIBDs, usually among those with small values of \( \lambda \).
Here we present results based on defining sets of full designs, that is, designs comprising all \( k \)-tuples on a given set of \( v \) elements. These defining sets are useful, despite their relatively large \( \lambda \) values, since we show that a defining set of any simple BIBD can often be derived from a defining set of the corresponding full design. This leads to an upper bound on the number of simple designs with given parameters, provided that a \( (v,k,\lambda) \) BIBD exists for minimum feasible \( \lambda \).
A backtracking over near parallel classes with an early isomorph rejection is carried out to enumerate all the near resolvable \( (2k+1, k, k-1) \) balanced incomplete block designs for \( 3 \leq k \leq 13 \). We first prove some results which enable us to restrict the search space of near parallel classes. The number of nonisomorphic designs is equal to 1 for each \( 3 \leq k \leq 8 \) and there are respectively 2, 0, 19, 8, and 374 nonisomorphic designs for \( k = 9, 10, 11, 12 \), and \( 13 \).
Let \( \gamma_t(G) \) denote the total domination number of the graph \( G \). A graph \( G \) is said to be total domination edge critical, or simply \( \gamma_t \)-critical, if \( \gamma_t(G+e) < \gamma_t(G) \) for each edge \( e \in E(\overline{G}) \). We show that, for \( 4_t \)-critical graphs \( G \), that is, \( \gamma_t \)-critical graphs with \( \gamma_t(G) = 4 \), the diameter of \( G \) is either \( 2 \), \( 3 \), or \( 4 \). Further, we characterize structurally the \( 4_t \)-critical graphs \( G \) with \( \text{diam}(G) = 4 \).
In the Shamir \( (k,n) \) threshold scheme, if one or more of the \( n \) shares are fake, then the secret may not be reconstructed correctly by some sets of \( k \) shares. Supposing that at most \( t \) of the \( n \) shares are fake, Rees et al. (1999) described two algorithms to determine consistent sets of shares so that the secret can be reconstructed correctly from \( k \) shares in any of these consistent sets. In their algorithms, no honest participant can be absent and at least \( n – t \) shares should be pooled during the secret reconstruction phase. In this paper, we propose a modified algorithm for this problem so that the number of participants taking part in the secret reconstruction can be reduced to \( k + 2t \) and the shares need to be pooled can be reduced to, in the best case, \( k + t \), and less than or equal to \( k + 2t \) in the others. Its performance is evaluated. A construction for \( t \)-coverings, which play key roles in these algorithms, is also provided.
A linear \( k \)-forest is a graph whose components are paths with lengths at most \( k \). The minimum number of linear \( k \)-forests needed to decompose a graph \( G \) is the linear \( k \)-arboricity of \( G \) and is denoted by \( la_k(G) \). In this paper, we study the linear \( 3 \)-arboricity of balanced complete multipartite graphs and we obtain some substantial results.
Let \( m_2(N, q) \) denote the size of the largest caps in \( PG(N, q) \) and let \( m_2′(N, q) \) denote the size of the second-largest complete caps in \( PG(N, q) \). Presently, it is known that \( m_2(4, 5) \leq 111 \) and that \( m_2(4, 7) \leq 316 \). Via computer searches for caps in \( PG(4, 5) \) using the result of Abatangelo, Larato, and Korchmáros that \( m_2′(3, 5) = 20 \), we improve the first upper bound to \( m_2(4, 5) \leq 88 \). Computer searches in \( PG(3, 7) \) show that \( m_2′(3, 7) = 32 \), and this latter result then improves the upper bound on \( m_2(4, 7) \) to \( m_2(4, 7) \leq 238 \). We also present the known upper bounds on \( m_2(N, 5) \) and \( m_2(N, 7) \) for \( N > 4 \).
A sum of disjoint products (SDP) representation of a Boolean function is useful because it provides readily available information about the function; however, a typical SDP contains many more terms than an equivalent ordinary sum of products. We conjecture the existence of certain particular SDP forms of \( x_1 + \cdots + x_t \), which could be used as patterns in creating relatively economical SDP forms of other Boolean functions.
In Algebraic Graph Theory, Biggs \([2]\) gives a method for finding the chromatic polynomial of any connected graph by computing the Tutte polynomial. It is used by Biggs \([2]\) to compute the chromatic polynomial of Peterson’s graph. In \(1972\) Sands \([4]\) developed a computer algorithm using matrix operations on the incidence matrix to compute the Tutte Polynomial. In \([1]\), Anthony finds the worst-case time-complexity of computing the Tutte Polynomial. This paper shows a method using group-theoretical properties to compute the Tutte polynomial for Cayley graphs which improves the time-complexity.
Let \(N({Z})\) denote the set of all positive integers (integers). The sum graph \(G_S\) of a finite subset \(S \subset N({Z})\) is the graph \((S, E)\) with \(uv \in E\) if and only if \(u+v \in S\). A graph \(G\) is said to be an (integral) sum graph if it is isomorphic to the sum graph of some \(S \subset N({Z})\). The (integral) sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which when added to \(G\) result in an (integral) sum graph. A mod (integral) sum graph is a sum graph with \(S \subset {Z}_m \setminus \{0\}\) (\(S \subset {Z}_m\)) and all arithmetic performed modulo \(m\) where \(m \geq |S|+1\) (\(m \geq |S|\)). The mod (integral) sum number \(\rho(G)\) of \(G\) is the least number \(\rho\) (\(\psi\)) of isolated vertices \(\rho K_1\) (\(\psi K_1\)) such that \(G \cup \rho K_1\) (\(G \cup \psi K_1\)) is a mod (integral) sum graph. In this paper, the mod (integral) sum numbers of \(K_{r,s}\) and \(K_n – E(K_r)\) are investigated and bounded, and \(n\)-spoked wheel \(W_n\) is shown to be a mod integral sum graph.
In this paper, the forcing domination numbers of the graphs \(P_n \times P_3\) and \(C_n \times P_3\) are completely determined. This improves the previous results on the forcing domination numbers of \(P_n \times P_2\) and \(C_n \times P_2\).
The stabilizers of the minimum-weight codewords of the binary codes obtained from the strongly regular graphs \(T(n)\) defined by the primitive rank-\(3\) action of the alternating groups \(A_n\), where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\) are examined. For a codeword \(w\) of minimum-weight in the binary code \(C\) obtained as stated above, from an adjacency matrix of the triangular graph \(T(n)\) defined by the primitive rank-3 action of the alternating groups \(A_n\) where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\), we determine the stabilizer \(Aut(C)_w\) in \(Aut(C)\) and show that \(Aut(C)_w\) is a maximal subgroup of \(Aut(C)\).
For a graph \(G\), let \(\mathcal{D}(G)\) be the set of strong orientations of \(G\). Define \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\) and \(\rho(G) = \overrightarrow{d}(G) – d(G)\), where \(d(D)\) (resp. \(d(G)\)) denotes the diameter of the digraph \(D\) (resp. graph \(G\)). In this paper, we determine the exact value of \(\rho(K_r \times K_s)\) for \(r \leq s\) and \((r,s) \not\in \{(3,5), (3,6), (4,4)\}\), where \(K_r \times K_s\) denotes the tensor product of \(K_r\) and \(K_s\). Using the results obtained here, a known result on \(\rho(G)\), where \(G\) is a regular complete multipartite graph is deduced as corollary.
A two-step approach to finding knight covers for an \(N \times N\) chessboard eliminates the problem of detecting duplicate partial solutions. The time and storage needed to generate solutions is greatly reduced. The method can handle boards as large as \(45 \times 45\) and has matched or beaten all previously known solutions for every board size tried.
In this paper we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-1}{2} \leq m \leq \frac{n^2-n}{2}\), when \(n\) is odd. Moreover, when \(n\) is even we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-n}{2}-(n-2) \leq m \leq \frac{n^2-n}{2}\) and \(m \in \{\frac{n^2}{4}, \frac{n^2}{4}+2, \frac{n^2}{4}+4, \ldots, \frac{n^2-n}{2}-n\}\).
In this paper, a characterization of two classes of \((q, q+1)\)-geometries, that are fully embedded in a projective space \(PG(n, q)\), is obtained. The first class is the one of the \((q,q+1)\)-geometry \(H^{n,m}_q\), having points the points of \(PG(n, q)\) that are not contained in an \(m\)-dimensional subspace \(\Pi[m]\) of \(PG(n, q)\), for \(0 \leq m \leq n-3\), and lines the lines of \(PG(n, q)\) skew to \(\Pi[m]\). The second class is the one of the \((q,q+1)\)-geometry \(SH^{n,m}_q\), having the same point set as \(H^{n,m}_q\), but with \(-1 \leq m \leq n-3\), and lines the lines skew to \(\Pi^{n,m}_q\) that are not contained in a certain partition of the point set of \(SH^{n,m}_q\). Our characterization uses the axiom of Pasch, which is also known as axiom of Veblen-Young. It is a generalization of the characterization for partial geometries satisfying the axiom of Pasch by J. A. Thas and F. De Clerck. A characterization for \(H^{n,m}_q\) was already proved by H. Cuypers. His result however does not include \(SH^{n,m}_q\).
In this note we construct nested partially balanced incomplete block designs based on \(NC_{m}\)-scheme. Secondly we construct NPBIB designs from a given PBIB design with \(\lambda_{1} = 1\) and \(\lambda_{2} = 0\) with same association scheme for both systems of PBIB designs. Finally, we give some results and examples where the two systems of PBIB designs in NPBIB designs have different association schemes.
This paper discusses the covering property and the Uniqueness Property of Minima (UPM) for linear forms in an arbitrary number of variables, with emphasis on the case of three variables (triple loop graph). It also studies the diameter of some families of undirected chordal ring graphs. We focus upon maximizing the number of vertices in the graph for given diameter and degree. We study the result in \([2]\), we find that the family of triple loop graphs of the form \(G(4k^2+2k+1; 1;2k+1; 2k^2)\) has a larger number of nodes for diameter \(k\) than the family \(G(3k^2+3k+1;1;3k+1;3k+2)\) given in \([2]\). Moreover, we show that both families have the Uniqueness Property of Minima.
In this paper, an algorithm based on. trades is presented to classify two classes of large sets of \(t\)-designs, namely \(LS[14](2, 5, 10)\) and \(LS[6](3, 5, 12)\).
In this work, we study which tubular surfaces verify that the embeddings of infinite, locally finite connected graphs without vertex accumulation points are embeddings without edge accumulation points. Furthermore, we characterize the graphs which admit embeddings with no edge accumulation points in the sphere with \(n\) ends in terms of forbidden subgraphs.
In this paper, self-centered, bi-eccentric splitting graphs are characterized. Further various bounds for domination number, global domination number and the neighborhood number of these graphs are obtained.
In this study we are going to give a new \((t,k)\)-geodetic set definition. This is a refinement of the geodetic set definition given in \([11]\). With this new definition we obtain more information about the graph. We also give a relationship between the \((t,k)\)-geodetic set and the integrity of a graph. By using a \((t,k)\)-geodetic set we give a new proof for the upper bound of integrity of trees and unicycle graphs.
For a long time we had thought that there does not exist an OGDD of type \(4^4\). In this article, an OGDD of type \(4^4\) will be constructed.
Consider a tree \(T = (V, E)\) with root \(r \in V\) and \(|V| = N\). Let \(p_v\) be the probability that a user wants to access node \(v\). A bookmark is an additional link from \(r\) to any other node of \(T\). We want to add \(k\) bookmarks to \(T\), so as to minimize the expected access cost from \(r\), measured by the average length of the shortest path. We present a characterization of an optimal assignment of \(k\) bookmarks in a perfect binary tree with uniform probability distribution of access and \(k \leq \sqrt{N + 1}\).
We show various combinatorial identities that are generated by tree counting arguments. In particular, we give formulas for \(n^p\) and \(\tau(K_{s,t})\) which establishes an equivalence.
The question of necessary and sufficient conditions for the existence of a simple \(3\)-uniform hypergraph with a given degree sequence is a long outstanding open question. We provide a result on degree sequences of \(3\)-hypergraphs which shows that any two \(3\)-hypergraphs with the same degree sequence can be transformed into each other using a sequence of trades, also known as null-\(3\)-hypergraphs. This result is similar to the Havel-Hakimi theorem for degree sequences of graphs.
In this paper we consider the problem as follows: Given a bipartite graph \(G = (V_1, V_2; E)\) with \(|V_1| = |V_2| = n\) and a positive integer \(k\), what degree condition is sufficient to ensure that for any \(k\) distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\), \(G\) contains \(k\) independent quadrilaterals \(Q_1, Q_2, \ldots, Q_k\) such that \(v_i \in V(Q_i)\) for every \(i \in \{1, 2, \ldots, k\}\), or \(G\) has a \(2\)-factor with \(k\) independent cycles of specified lengths with respect to \(\{v_1, v_2, \ldots, v_k\}\)? We will prove that if \(d(x) + d(y) \geq \left\lceil (4n + k)/3 \right\rceil\) for each pair of nonadjacent vertices \(x \in V_1\) and \(y \in V_2\), then, for any \(k\) distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\), \(G\) contains \(k\) independent quadrilaterals \(Q_1, Q_2, \ldots, Q_k\) such that \(v_i \in V(Q_i)\) for each \(i \in \{1, \ldots, k\}\). Moreover, \(G\) has a \(2\)-factor with \(k\) cycles with respect to \(\{v_1, v_2, \ldots, v_k\}\) such that \(k – 1\) of them are quadrilaterals. We also discuss the degree conditions in the above results.
We call the graph \(G\) an edge \(m\)-coloured if its edges are coloured with \(m\) colours. A path (or a cycle) is called monochromatic if all its edges are coloured alike. A subset \(S \subseteq V(G)\) is independent by monochromatic paths if for every pair of different vertices from \(S\) there is no monochromatic path between them. In \([5]\) it was defined the Fibonacci number of a graph to be the number of all independent sets of \(G\); recall that \(S\) is independent if no two of its vertices are adjacent. In this paper we define the concept of a monochromatic Fibonacci number of a graph which gives the total number of monochromatic independent sets of \(G\). Moreover we give the number of all independent by monochromatic paths sets of generalized lexicographic product of graphs using the concept of a monochromatic Fibonacci polynomial of a graph. These results generalize the Fibonacci number of a graph and the Fibonacci polynomial of a graph.
Let \(D = (V, E)\) be a primitive digraph. The exponent of \(D\) at a vertex \(u \in V\), denoted by \(\exp_D(u)\), is defined to be the least integer \(k\) such that there is a walk of length \(k\) from \(u\) to \(v\) for each \(v \in V\). Let \(V = \{v_1, v_2, \ldots, v_n\}\). The vertices of \(V\) can be ordered so that \(\exp_D(v_{i_1}) \leq \exp_D(v_{i_2}) \leq \ldots \leq \exp_D(v_{i_n}) = \gamma(D)\). The number \(\exp_p(v_n)\) is called the \(k\)-exponent of \(D\), denoted by \(\exp_p(k)\). We use \(L(D)\) to denote the set of distinct lengths of the cycles of \(D\). In this paper, we completely determine the \(1\)-exponent sets of primitive, minimally strong digraphs with \(n\) vertices and \(L(D) = \{p, q\}\), where \(3 \le p < q\) and \(p + q > n\).
Let \(\mathcal{C}\) be any class of finite graphs. A graph \(G\) is \(\mathcal{C}\)-ultrahomogeneous if every isomorphism between induced subgraphs belonging to \(\mathcal{C}\) extends to an automorphism of \(G\). We study finite graphs that are \({K}_*\)-ultrahomogeneous, where \({K}_*\) is the class of complete graphs. We also explicitly classify the finite graphs that are \(\sqcup{K}_{*}\)-ultrahomogeneous, where \(\sqcup{K}_{*}\) is the class of disjoint unions of complete graphs.
For any positive integer \(n\), let \(S_n\), denote the set of all permutations of the set \(\{1,2,\ldots,n\}\). We think of a permutation just as an ordered list. For any \(p\) in \(S_n\), and for any \(i \leq n\), let \(p \downarrow i\) be the permutation on the set \(\{1,2,\ldots,n – 1\}\) obtained from \(p\) as follows: delete \(i\) from \(p\) and then subtract \(1\) in place from each of the remaining entries of \(p\) which are larger than \(i\). For any \(p\) in \(S_n\), we let \(R(p) = \{q \in S_{n-1} : g = p \downarrow i \;\text{for some} \;i \leq n\}\), the set of reductions of \(p\). It is shown that, for \(n > 4\), any \(p\) in \(S_n\), is determined by its set of reductions \(R(p)\).
For \( n \in \mathbb{N} \), we interpret the vertex set \( W_n \) of the \( n \)-cube as a vector space over the field \( \mathbb{F}_2 \) and prove that a regular \( n \)-simplex can be inscribed into the \( n \)-cube such that its vertices constitute a subgroup of \( W_n \) if and only if \( n+1 \) is a power of 2. Furthermore, a connection to the theory of Hamming Codes will be established.
An \( (n,k) \) binary self-orthogonal code is an \( (n,k) \) binary linear code \( C \) that is contained in its orthogonal complement \( C^\bot \). A self-orthogonal code \( C \) is self-dual if \( C = C^\bot \). Two codes, \( C_1 \) and \( C_2 \), are \({equivalent}\) if and only if there exists a coordinate permutation of \( C_1 \) that takes \( C_1 \) into \( C_2 \). The automorphism group of a code \( C \) is the set of all coordinate permutations of \( C \) that takes \( C \) into itself.
This paper is a continuation of the work presented in [2], in which we described an algorithm for enumerating inequivalent binary self-dual codes. We used our algorithm to enumerate the self-dual codes of length up to and including 32. Our algorithm also found the size of the automorphism group of each code.
We have since made several improvements to our algorithm. It now generally runs faster. It also now finds generators for the automorphism group of each code. We have used our improved algorithm to enumerate the self-dual codes of length 34. We have also found the automorphism groups for each of our self-dual codes of length less than or equal to 34. The list of length 34 codes are new, as are the lists of automorphism groups for the length 32 and length 34 codes. We have found there are 19914 inequivalent length 34 codes with distance 4 and 938 length 34 codes with distance 6.
A graph is claw-free if it has no induced \( K_{1,3} \) subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edge-connected claw-free graph has a spanning Eulerian subgraph with maximum degree at most 4.
A labeling \( f \) of a graph \( G \) is called semi-H-cordial if for each vertex \( v \), \( |f(v)| \leq 1 \), \( |e_f(1) – e_f(-1)| \leq 1 \) and \( |v_f(1) – v_f(-1)| \leq 1 \). In this paper we study the forcing semi-H-cordial numbers of paths, cycles, stars, trees, Dutch-windmill graphs, wheels, grids and cylinders.
A three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) is a three-fold resolvable packing with maximum possible number of parallel classes, each containing one block of size 3 and all other blocks of size 4. This article investigates the spectra of three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) for \( s = 5 \) and \( 6 \), and we show that it contains all positive integers \( v \equiv s – 4 \pmod{4} \) with \( v \geq 17 \) if \( s = 5 \), and \( v \geq 26 \) if \( s = 6 \).
Let \( G \) be a \( (p,q) \)-graph in which the edges are labeled \( 1, 2, 3, \ldots, q \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If \( G \) can be labeled so that the vertex sums are distinct, mod \( p \), then \( G \) is said to be edge-graceful. If the edges of \( G \) can be labeled \( 1, 2, 3, \ldots, q \) so that the vertex sums are constant, mod \( p \), then \( G \) is said to be edge-magic. It is conjectured by Lee [9] that any connected simple \( (p,q) \)-graph with \( q(q+1) \equiv p(p-1)/2 \pmod{p} \) vertices is edge-graceful. We show that the conjecture is true for maximal outerplanar graphs. We also completely determine the edge-magic maximal outerplanar graphs.
We enumerate the self-orthogonal Latin squares of orders \(1\) through \(9\) and discuss the nature of the isomorphism classes of each order. Furthermore, we consider the possibility of enlarging sets of self-orthogonal Latin squares to produce complete sets.
A vertex-magic total labeling on a graph with \( v \) vertices and \( e \) edges is a one-to-one map taking the vertices and edges onto the integers \( 1, 2, \ldots, v+e \) with the property that the sum of the label on a vertex and the labels of its incident edges is constant, independent of the choice of vertex. We give vertex-magic total labelings for several classes of regular graphs. The paper concludes with several conjectures and open problems in the area.
In this paper, we complete the classification of optimal binary linear self-orthogonal codes up to length 25. Optimal self-orthogonal codes are also classified for parameters up to length 40 and dimension 10. The results were obtained via two independent computer searches.
In this paper we examine the classical Williamson construction for Hadamard matrices, from the point of view of a striking analogy with isomorphisms of division algebras. By interpreting the 4 Williamson array as a matrix arising from the real quaternion division algebra, we construct Williamson arrays with 8 matrices, based on the real octonion division algebra. Using a Computational Algebra formalism we perform exhaustive searches for even-order 4-Williamson matrices up to 18 and odd- and even-order 8-Williamson matrices up to 9 and partial searches for even-order 4-Williamson matrices up to 22 and odd- and even-order 8-Williamson matrices for orders 10 — 13. Using Magma, we conduct searches for inequivalent Hadamard matrices within all the sets of matrices obtained by exhaustive and partial searches. In particular, we establish constructively ten new lower bounds for the number of inequivalent, Hadamard matrices of the consecutive orders 72, 76, 80, 84, 88, 92, 96, 100, 104 and 108.
This article continues the study of a class of non-terminating expansions of sin\( (m\alpha) \) (even \(m \geq 2 \)) which in each case possesses embedded Catalan numbers. A known series form of the sine function (said to be associated with Euler) is taken here as our basic representation, the coefficient of the general term being developed analytically in an interesting fashion and shown to be dependent on the Catalan sequence in the manner expected.
The work, which has a historical backdrop to it, is discussed in the context of prior results by the author and others.
A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity \(\lambda’\) of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each connected component has at least two vertices. A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \min\{d_G(u)+d_G(v)-2: uv \text{ is an edge in } G\}\). This paper proves that for any \(d\) and \(n\) with \(d \geq 2\) and \(n\geq 1\) the Kautz undirected graph \(UK(d, 1)\) is \(\lambda’\)-optimal except \(UK(2,1)\) and \(UK(2,2)\) and, hence, is super edge-connected except \(UK(2, 2)\).
In a \((k, d)\)-relaxed coloring game, two players, Alice and Bob, take turns coloring the vertices of a graph \(G\) with colors from a set \(C\) of \(k\) colors. A color \(c\) is legal for an uncolored vertex (at a certain step) means that after coloring \(x\) with color \(i\), the subgraph induced by vertices of color \(i\) has maximum degree at most \(d\). Each player can only color a vertex with a legal color. Alice’s goal is to have all the vertices colored, and Bob’s goal is the opposite: to have an uncolored vertex without legal color. The \(d\)-relaxed game chromatic number of a graph \(G\), denoted by \(\chi^{(d)}_g(G)\) is the least number \(k\) so that when playing the \((k,d)\)-relaxed coloring game on \(G\), Alice has a winning strategy. This paper proves that if \(G\) is an outer planar graph, then \(\chi^{(d)}_g(G) \leq 7 – d\) for \(d = 0, 1, 2, 3, 4\).
A graph \(G\) is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let \(X\) be a 2-stratified graph with one fixed blue vertex \(v\) specified. We say that \(X\) is rooted at \(v\). The \(X\)-domination number of a graph \(G\) is the minimum number of red vertices of \(G\) in a red-blue coloring of the vertices of \(G\) such that every blue vertex \(uv\) of \(G\) belongs to a copy of \(X\) rooted at \(v\). In this paper we investigate the \(X\)-domination number of prisms when \(X\) is a 2-stratified 4-cycle rooted at a blue vertex.
The maximum possible volume of a simple, non-Steiner \((v, 3, 2)\) trade was determined for all \(v\) by Khosrovshahi and Torabi (Ars Combinatoria \(51 (1999), 211-223)\); except that in the case \(v \equiv 5\) (mod 6), \(v \geq 23\), they were only able to provide an upper bound on the volume. In this paper we construct trades with volume equal to that bound for all \(v \equiv 5\) (mod 6), thus completing the problem.
For a positive integer \(n\), let \(G\) be \(K_n\) if \(n\) is odd and \(K_n\) less a one-factor if \(n\) is even. In this paper it is shown that, for non-negative integers \(p\), \(q\) and \(r\), there is a decomposition of \(G\) into \(p\) \(4\)-cycles, \(q\) \(6\)-cycles and \(r\) \(8\)-cycles if \(4p + 6q + 8r = |E(G)|\), \(q = 0\) if \(n < 6\), and \(r = 0\) if \(n < 8\).
This paper introduces a bijection between RNA secondary structures and bicoloured ordered trees.
For a given triangle \(T\), consider the problem of finding a finite set \(S\) in the plane such that every two-coloring of \(S\) results in a monochromatic set congruent to the vertices of \(T\). We show that such a set \(S\) must have at least seven points. Furthermore, we show by an example that the minimum of seven is achieved.
The two games considered are mixtures of Searching and Cops and Robber. The cops have partial information, provided first via selected vertices of a graph, and then via selected edges. This partial information includes the robber’s position, but not the direction in which he is moving. The robber has perfect information. In both cases, we give bounds on the amount of such information required by a single cop to guarantee the capture of the robber on a cop-win graph.
Let \(K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-design of \(K_v\), denoted by \(G\)-GD(\(v\)), is a pair of (\(X\), \(\mathcal{B}\)), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, the discussed graphs are sixteen graphs with six vertices and seven edges. We give a unified method for constructing such \(G\)-designs.
Graceful labellings have both a mathematical beauty in their own right and considerable connections with pure and applied combinatorics (edge-decomposition of graphs, coding systems, communication networks, etc.). In the present paper, we exhibit a graceful labelling for each generalized Petersen graph \(P_{8t,3}\) with \(t \geq 1\). As a consequence, we obtain, for any fixed \(t\), a cyclic edge-decomposition of the complete graph \(K_{48t+1}\) into copies of \(P_{8t,3}\). Due to its extreme versatility, the technique employed looks promising for finding new graceful labellings, not necessarily involving generalized Petersen graphs.
A graph on \(n\) vertices having no vertex of degree greater than \(f\), \(2 \leq f \leq n – 2\), is called an \(f\)-graph of order \(n\). For a given \(f\), the vertices of degree less than \(f\) are called orexic. An \(f\)-graph to which no edge can be added without violating the \(f\)-degree restriction is called an edge maximal \(f\)-graph (EM \(f\)-graph). An upper bound, as a function of \(n\) and \(f\), for the number of orexic vertices in an EM \(f\)-graph and the structure of the subgraph induced by its orexic vertices is given. For any \(n\) and \(f\), the maximum size, minimum size, and realizations of extremal size EM \(f\)-graphs having \(m\) orexic vertices and order \(n\) are obtained. This is also done for any given \(n\) and \(f\) independent of \(m\). The number of size classes of EM \(f\)-graphs of order \(n\) and fixed \(m\) is determined. From this, the maximum number of size classes over all \(m\) follows. These results are related to the study of \((f + 1)\)-star-saturated graphs.
We give a decomposition formula for the edge zeta function of a regular covering \(\overrightarrow{G}\) of a graph \(G\). Furthermore, we present a determinant expression for some \(Z\)-function of an oriented line graph \(\overrightarrow{L}(G)\) of \(G\). As a corollary, we obtain a factorization formula for the edge zeta function of \(\overrightarrow{G}\) by \(L\)-functions of \(\overrightarrow{L}(G)\).
A hamiltonian graph \(G\) is panpositionable if for any two different vertices \(x\) and \(y\) of \(G\) and any integer \(k\) with \(d_G(x,y) \leq k \leq |V(G)|/2\), there exists a hamiltonian cycle \(C\) of \(G\) with \(d_C(x,y) = k\). A bipartite hamiltonian graph \(G\) is bipanpositionable if for any two different vertices \(x\) and \(y\) of \(G\) and for any integer \(k\) with \(d_G(x,y) \leq k \leq |V(G)|/2\) and \((k – d_G(x,y))\) is even, there exists a hamiltonian cycle \(C\) of \(G\) such that \(d_C(x,y) = k\). In this paper, we prove that the hypercube \(Q_n\) is bipanpositionable hamiltonian if and only if \(n \geq 2\). The recursive circulant graph \(G(n;1,3)\) is bipanpositionable hamiltonian if and only if \(n \geq 6\) and \(n\) is even; \(G(n; 1,2)\) is panpositionable hamiltonian if and only if \(n \in \{5,6,7,8,9, 11\}\), and \(G(n; 1, 2,3)\) is panpositionable hamiltonian if and only if \(n \geq 5\).
The lower domination number of a digraph \(D\), denoted by \(\gamma(D)\), is the least number of vertices in a set \(S\), such that \(O[S] = V(D)\). A set \(S\) is irredundant if for all \(x \in S\), \(|O[x] – O[S – x]| \geq 1\). The lower irredundance number of a digraph, denoted \(ir(D)\), is the least number of vertices in a maximal irredundant set. A Gallai-type theorem has the form \(x(G) + y(G) = n\), where \(x\) and \(y\) are parameters defined on \(G\), and \(n\) is the number of vertices in the graph. We characterize directed trees satisfying \(\gamma(D) + \Delta_+(D) = n\) and directed trees satisfying \(ir(D) + \Delta_+(D) = n\).
We introduce a new concept of strong domination and connected strong domination in hypergraphs. The relationships between strong domination number and other hypergraph parameters like domination, independence, strong independence and irredundant numbers of hypergraphs are considered. There are also some chains of inequalities generalizing the famous Cockayne, Hedetniemi and Miller chain for parameters of graphs. There are given some generalizations of well known theorems for graphs, namely Gallai type theorem generalizing Nieminen, Hedetniemi and Laskar theorems.
The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. In this paper, we seek to convert vertex linear arboricity into its fractional analogues, i.e., the fractional vertex linear arboricity of graphs. Let \(\mathbb{Z}_n\) denote the additive group of integers modulo \(n\). Suppose that \(C \subseteq \mathbb{Z}_n \backslash 0\) has the additional property that it is closed under additive inverse, that is, \(-c \in C\) if and only if \(c \in C\). A circulant graph is the graph \(G(\mathbb{Z}_n, C)\) with the vertex set \(\mathbb{Z}_n\) and \(i, j\) are adjacent if and only if \(i – j \in C\). The fractional vertex linear arboricity of the complete \(n\)-partite graph, the cycle \(C_n\), the integer distance graph \(G(D)\) for \(D = \{1, 2, \ldots, m\}\), \(D = \{2, 3, \ldots, m\}\) and \(D = P\) the set of all prime numbers, the Petersen graph and the circulant graph \(G(\mathbb{Z}_a, C)\) with \(C = \{-a + b, \ldots, -b, b, \ldots, a – b\}\) (\(a – 2b \geq b – 3 \geq 3\)) are determined, and an upper and a lower bounds of the fractional vertex linear arboricity of Mycielski graph are obtained.
Deciding whether a graph can be partitioned into \(k\) vertex-disjoint paths is a well-known NP-complete problem. In this paper, we give new sufficient conditions for a bipartite graph to be partitionable into \(k\) vertex-disjoint paths. We prove the following results for a simple bipartite graph \(G = (V_1, V_2, E)\) of order \(n\):(i) For any positive integer \(k\), if \(\|V_1| – |V_2\| \leq k\) and \(d_G(x) + d_G(y) \geq \frac{n-k+1}{2}\) for every pair \(x \in V_1\) and \(y \in V_2\) of nonadjacent vertices of \(G\), then \(G\) can be partitioned into \(k\) vertex-disjoint paths, unless \(k = 1\), \(|V_1| = |V_2| = \frac{n}{2}\) and \(G = K_{s,s} \cup K_{\frac{n}{2} – s, \frac{n}{2} – s} \cup K_{2, 2}\), where \(1 \leq k \leq \frac{n}{2} – 1\);(ii) For any two positive integers \(p_1\) and \(p_2\) satisfying \(n = p_1 + p_2\), if \(G\) does not belong to some easily recognizable classes of exceptional graphs, \(\|V_1| – |V_2\| \leq 2\) and \(d_G(x) + d_G(y) = \frac{n-1}{2}\) for every pair \(x \in V_1\) and \(y \in V_2\) of nonadjacent vertices of \(G\), then \(G\) can be partitioned into two vertex-disjoint paths \(P_{1}\) and \(P_{2}\) of order \(p_1\) and \(p_2\), respectively.These results also lead to new sufficient conditions for the existence of a Hamilton path in a bipartite graph.
In this paper we compute the chirality group, the chirality index and the smallest regular coverings of the chiral Coxeter maps, the toroidal orientably regular maps described in Coxeter and Moser monograph [H.S.M.Coxeter and W.O.J.Moser,Generation and Relations Discrete Group(4th ed.),Springer-varlag,Berlin,1984]. We also compute the greatest regular maps covered by chiral Coxeter maps.
We observe that a lobster with diameter at least five has a unique path \(x_0, x_1, \ldots, x_{m}\) (called the central path) such that \(x_p\) and \(x_m\) are adjacent to the centers of at least one \(K_{1,s}\), \(s > 0\), and besides adjacencies in the central path each \(x_i\), \(1 \leq i \leq m-1\), is at most adjacent to the centers of some \(K_{1,s}\), \(s \geq 0\). In this paper we give graceful labelings to some new classes of lobsters with diameter at least five, in which the degree of the vertex \(x_m\) is odd and the degree of each of the remaining vertices on the central path is even. The main idea used to obtain these graceful lobsters is to form a diameter four tree \(T(L)\) from a lobster \(L\) of certain type, give a graceful labeling to \(T(L)\) and finally get a graceful labeling of \(L\) by applying component moving and inverse transformations.
A graph \(G = (V, E)\) is said to be super edge-magic if there exists a one-to-one correspondence \(A\) from \(V \cup E\) onto \(\{1, 2, 3, \ldots, |V| + |E|\}\) such that \(\lambda(V) = \{1, 2, \ldots, |V|\}\) and \(\lambda(x) + \lambda(xy) + \lambda(y)\) is constant for every edge \(xy\).In this paper, given a positive integer \(k\) (\(k \geq 6\)), we use the partitions of \(k\) having three distinct parts to construct infinitely many super edge-magic graphs without isolated vertices with edge magic number \(k\). Especially, we use this method to find graphs with the maximum number of edges among the super edge-magic graphs with \(v\) vertices. In addition, we investigate whether or not some interesting families of graphs are super edge-magic.
A vertex \(w\) in a di(graph) \(G\) is said to resolve a pair \(u, v\) of vertices of \(G\) if the distance from \(u\) to \(w\) does not equal the distance from \(v\) to \(w\). A set \(S\) of vertices of \(G\) is a resolving set for \(G\) if every pair of vertices of \(G\) is resolved by some vertex of \(S\). The smallest cardinality of a resolving set for \(G\), denoted by \(dim(G)\), is called the metric dimension for \(G\).
We show that if \(G\) is the Cayley digraph \(Cay(\Delta : \Gamma)\) where \(\Delta = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}\) and \(\Gamma =\mathbb{Z}_m \oplus \mathbb{Z}_n \oplus \mathbb{Z}_k\) with \(m \leq n \leq k\), then \(dim(G) = n\) if \(m < n\) and improve known upper bounds if \(m = n\). We use these results to establish improved upper bounds for the metric dimension of Cayley digraphs of abelian groups that are expressed as a direct product of four or more cyclic groups. Lower bounds for Cayley digraphs of groups that are multiple copies of \(\mathbb{Z}_n\) are established.
In this paper, the unimodality of \((r,\beta)\)-Stirling numbers and certain asymptotic approximation of \((r,\beta)\)-Bell numbers are established. Together with these results and the most general form of Central Limit Theorem, viz. Bounded Variance Normal Convergence Criterion, the \((r,\beta)\)-Stirling numbers are shown to be asymptotically normal.
The graph \(\mathcal{R}(d)\) of realizations of \(d\) is a graph whose vertices are the graphs with degree sequence \(d\), two vertices are adjacent in the graph \(\mathcal{R}(d)\) if one can be obtained from the other by a switching. It has been shown that the graph \(\mathcal{R}(d)\) is connected. Let \(\mathcal{CR}(d)\) be the set of connected graphs with degree sequence \(d\). Taylor \([13]\) proved that the subgraph of \(\mathcal{R}(d)\) induced by \(\mathcal{CR}(d)\) is connected. Several connected subgraphs of \(\mathcal{CR}(d)(3^n)\) are obtained in this paper. As an application, we are able to obtain the interpolation and extremal results for the number of maximum induced forests in the classes of connected subgraphs of \(\mathcal{CR}(d)(3^n)\).
Let \(T = (V, A)\) be a finite tournament with \(n \geq 2\) vertices. The dual of T is the tournament \(T^* = (V, A^*)\) defined by: for all \(x,y \in V, (x,y) \in A^*\) if and only if \((y,x) \in A\). The tournament \(T\) is critical if \(T\) is indecomposable and if for all \(x \in V\), the subtournament \(T(V – \{x\})\) is decomposable. A \(3\)-cycle is a tournament isomorphic to the tournament \(T, = ({0,1,2}, {(0, 1), (1, 2), (2, 0)})\). Let \(F\) be a set of non negative integers \(k < n\). The tournament \(T\) is \(F\)-selfdual if for every subset \(X\) of \(V\) such that \(|X |\in F\), the subtournaments \(T(X)\) and \(T^*(X)\) are isomorphic. In this paper, we study, for each integer \(k \geq 1\), the \(\{n – k\}\)-selfduality of the tournaments, with \(n \geq 4+k\) vertices, that are lexicographical sums of tournaments under a \(3\)-cycle or a critical tournament. As application, we determine for each integer \(k \geq 1\), the tournaments, with \(n \geq 4+ k\) vertices, that are \(\{4,n – k\}\)-selfdual.
This paper studies in detail the collection of closed sets of a matroid of arbitrary cardinality ordered by inclusion. The relation between the collection, in particular the collection of a simple matroid, and a finite length geometric lattice is dealt with. Finally, one obtains that up to isomorphism, a finite length geometric lattice is a simple matroid, and vice versa.
A vertex set \(D\) of a graph \(G\) is a dominating set if every vertex not in \(D\) is adjacent to some vertex in \(D\). The domination number \(\gamma\) of a graph \(G\) is the minimum cardinality of a dominating set in \(G\).
In 1975, Payan \([6]\) communicated without proof the inequality
\[2\gamma \leq {n} + 1 – \delta\]
for every connected graph not isomorphic to the complement of a one-regular graph, where \(n\) is the order and \(\delta\) the minimum degree of the graph. A first proof of (*) was published by Flach and Volkman \([3]\) in \(1980\).
In this paper, we firstly present a more transparent proof of (*). Using the idea of this proof, we show that
\[2\gamma \leq n – \delta\]
for connected graphs with exception of well-determined families of graphs.
A \((v,k,\lambda)\) covering design is a set of \(b\) blocks of size \(k\) such that each pair of points occurs in at least \(\lambda\) blocks, and the covering number \(C(v, k, \lambda)\) is the minimum value of \(b\) in any \((v, k, \lambda)\) covering design. For \(k = 5\) and \(v\) even, there are 24 open cases with \(2 \leq \lambda \leq 21\), each of which is the start of an open series for \(\lambda,\lambda + 20, \lambda + 40, \ldots\). In this article, we solve 22 of these cases with \(\lambda \leq 21\), leaving open \((v, 5, \lambda)=(44, 5, 13)\) and \((44, 5, 17)\) (and the series initiated for the former).
The basis number of a graph \( G \) is defined to be the least integer \( d \) such that there is a basis \( \mathcal{B} \) of the cycle space of \( G \) such that each edge of \( G \) is contained in at most \( d \) members of \( \mathcal{B} \). MacLane [16] proved that a graph, \( G \), is planar if and only if the basis number of \( G \) is less than or equal to 2. Ali and Marougi [3] proved that the basis number of the strong product of two cycles and a path with a star is less than or equal to 4. In this work, (1) we prove the basis number of the strong product of two cycles is 3. (2) We give the exact basis number of a path with a tree containing no subgraph isomorphic to a 3-special star of order 7. (3) We investigate the basis number of a cycle with a tree containing no subgraph isomorphic to a 3-special star of order 7. The results in (1) and (2) improve the upper bound of the basis number of the strong product of two cycles and a star with a path which were obtained by Ali and Marougi.
A set \( S \) of vertices is a total dominating set of a graph \( G \) if every vertex of \( G \) is adjacent to some vertex in \( S \). The minimum cardinality of a total dominating set is the total domination number \( \gamma_t(G) \). We show that for a nontrivial tree \( T \) of order \( n \) and with \( \ell \) leaves, \( \gamma_t(T) \geqslant \frac{n + 2 – \ell}{2} \), and we characterize the trees attaining this lower bound.
This paper presents a computationally efficient algorithm for solving the following well-known die problem: Consider a “crazy die” to be a die with \( n \) faces where each face has some “cost”. Costs need not be sequential. The problem is to determine the exact probability that the sum of costs from \( U \) throws of this die is \( \geq T \), \( T \in \mathbb{R} \). Our approach uses “slice” volume computation in \( U \)-dimensional space. Detailed algorithms, complexity analysis, and comparison with traditional generating functions approach are presented.
Difference systems of sets (DSS), introduced by Levenshtein, are used to design code synchronization in the presence of errors. The paper gives a new lower bound of DSS’s size.
In a loop transversal code, the set of errors is given the structure of a loop transversal to the linear code as a subgroup of the channel. A greedy algorithm for specifying the loop structure, and thus for the construction of loop transversal codes, was discussed by Hummer et al. Apart from some theoretical considerations, the focus was mainly on error correction, in the white noise case constructing codes with odd minimum distance. In this paper, an algorithm to compute loop transversal codes with even minimum distance is given. Some record-breaking codes over a 7-ary alphabet are presented.
Let \( a, b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:
\[
Q(a) =
\begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a+\frac{q-2}{2})\} & \text{if } q \text{ is even} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/{2})\} & \text{if } q \text{ is odd}
\end{cases}
\]
\[
P(b) =
\begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/{2})\} & \text{if } p \text{ is even} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (\frac{p-3}{2})/2)\} & \text{if } p \text{ is odd}
\end{cases}
\]
For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+ : V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and
\[
f^+(u) = \sum\{f(u,v) : (u,v) \in E(G)\}.
\]
We investigate \( Q(a)P(b) \) super edge-graceful graphs.
Let \( A \) be a non-trivial abelian group. We call a graph \( G = (V,E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A \setminus \{0\} \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum f(u,v) \) where the sum is over all \( (u,v) \in E(G) \), is a constant map. In this paper, we show that \( K_{k_1,k_2,\ldots,k_n} \) (where \( K_{i} \geq 2 \)) is \( A \)-magic, for all \( A \) where \( |A| \geq 3 \).
We define 1 new type of resolvability called \( \alpha \)-pair-resolvability in which each point appears in each resolution class as a member of \( \alpha \)-pairs. The concept is intended for path designs (or other designs) in which the role of points in blocks is not uniform or for designs which are not balanced. We determine the necessary conditions and show they are sufficient for \( k = 3 \) and \( \alpha = 2,3 \) (\( \alpha \geq 2 \) is necessary in every case). We also consider near \( a \)-pair-resolvability and show the necessary conditions are sufficient for \( \alpha = 2,4 \). We consider under what conditions it is possible for the ordered blocks of a path design to be considered as unordered blocks and thereby create a triple system (a tight embedding) and there also we show the necessary conditions are sufficient. We show it is always possible to embed maximally unbalanced path designs \( \text{PATH}(v, 3, 1) \) into \( \text{PATH}(v + s, 3, 1) \) for admissible \( s \), and to embed any \( \text{PATH}(v, 3, 2\lambda) \) into a \( \text{PATH}(v + s,3, 2\lambda) \) for any \( s \geq 1 \).
Recent developments in logic programming are based on bilattices (algebras with two separate lattice structures). This paper provides characterizations and structural descriptions for bilattices using the algebraic concepts of superproduct and hyperidentity. The main structural description subsumes the many variants that have appeared in the literature.
The Ramsey number \( R(C_4, B_n) \) is the smallest positive integer \( m \) such that for every graph \( F \) of order \( m \), either \( F \) contains \( C_4 \) (a quadrilateral) or \( \overline{F} \) contains \( B_n \) (a book graph \( K_2 + \overline{K_n} \) of order \( n+2 \)). Previously, we computed \( R(C_4, B_n) = n+9 \) for \( 8 \leq n \leq 12 \). In this continuing work, we find that \( R(C_4, B_{13}) = 22 \) and surprisingly \( R(C_4, B_{14}) = 24 \), showing that their values are not incremented by one, as one might have suspected. The results are based on computer algorithms.
Comma-free codes are used to correct synchronization errors in sequential transmission. Systematic comma-free codes have codewords with fixed positions for error correction. We consider only comma-free codes with constant word length \( n > 1 \). Circular codes use the integers mod \( n \) as indices for codeword entries. We first show two easily stated conditions are equivalent to the existence question for circular systematic comma-free codes over arbitrary finite alphabets. For \( n > 3 \) a family of circular systematic comma-free codes with word length \( n = p \), a prime, is constructed, each corresponding to a fair partition of a difference set in \( \mathbb{Z}_n \).
This paper gives the exact size of edit spheres of radius 1 and 2 for any word over a finite alphabet. Structural information about the edit metric space, in particular a representation as a pyramid of hypercubes, will be given. The 1-spheres are easy to understand, being identical to 1-spheres over the Hamming metric. Edit metric 2-spheres are much more complicated. The size of a 2-sphere hinges on the structure of the word at its center. That is, the word’s length, number of blocks, and most importantly (and troublesome) the number of locally maximal alternating substrings (LMAS) of each length. An alternating substring switches back and forth between two characters, e.g. 010101, and is maximal if it is contained in no other such substring. This variation in sphere size depending on center characteristics is what truly separates the algebraic character of codes over the edit metric from those over the Hamming metric.
We determine some coefficients of the flow polynomial of the complete graph \( K_n \).
Groups provide the mathematical language for exact symmetry. Applications in biology and other fields are now raising the problem of developing a rigorous theory of approximate symmetry. In this paper, it is shown how approximate symmetry is determined by a quasigroup.
A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices \(x\), the subgraph induced by the neighbour set of \(x\) is isomorphic to the subgraph induced by the closed non-neighbour set of \(x\). Graphs with the ncc property were characterized in [1] by the existence of a locally \(C_4\) perfect matching \(M\): every two edges of \(M\) induce a subgraph isomorphic to \(C_4\). In the present article, we investigate variants of locally \(C_4\) perfect matchings. We consider the cases where pairs of distinct edges of the matching induce isomorphism types including \(P_4\), the paw, or the diamond. We give several characterizations of graphs with such matchings. In addition, we supply characterizations of graphs with matchings whose edges satisfy a prescribed parity condition.
In this paper we obtain some necessary conditions for the existence of balanced arrays (B-arrays) with two symbols and having strength seven. We then describe how these conditions involving the parameters of the array can be used to obtain an upper bound on the constraints of such arrays, and give some illustrative examples to this effect.
A defensive alliance in a graph \( G(V,E) \) is a set of vertices \( S \subseteq V \) such that for every vertex \( v \in S \), the closed neighborhood \( N_G[v] \) of \( v \) has at least as many vertices in \( S \) as it has in \( V – S \). An offensive alliance is a set of vertices \( S \subseteq V \), such that for every vertex \( v \) in the boundary \( \partial(S) \) of \( S \) the number of neighbors that \( v \) has in \( S \) is greater than or equal to the number of neighbors it has in \( V – S \). A subset of vertices which is both an offensive and a defensive alliance is called a powerful alliance. An alliance which is also a dominating set is called a global alliance. In this paper, we show that finding an optimal defensive (offensive, powerful) global alliance is an NP-hard problem.
We discuss a branch of Ramsey theory concerning vertex Folkman numbers and how computer algorithms have been used to compute a new Folkman number. We write \( G \rightarrow (a_1, \ldots, a_k)^v \) if for every vertex \( k \)-coloring of an undirected simple graph \( G \), a monochromatic \( K_{a_i} \) is forced in color \( i \in \{1, \ldots, k\} \). The vertex Folkman number is defined as\(F_v(a_1, \ldots, a_k; p) = \text{min}\{|V(G)| : G \rightarrow (a_1, \ldots, a_k)^v \wedge K_p \nsubseteq G\}.\) Folkman showed in 1970 that this number exists for \( p > \text{max}\{a_1, \ldots, a_k\} \). Let \( m = 1 + \sum_{i=1}^k (a_i – 1) \) and \( a = \text{max}\{a_1, \ldots, a_k\} \), then \(F_v(a_1, \ldots, a_k; p) = m \text{ for } p > m,\) and \(F_v(a_1, \ldots, a_k; p) = a + m \text{ for } p = m.\)For \( p < m \) the situation is more difficult and much less is known. We show here that, for a case of \( p = m – 1 \), \( F_v(2, 2, 3; 4) = 14 \).
We classify all finite linear spaces on at most \(15\) points admitting a blocking set. There are no such spaces on \(11\) or fewer points, one on \(12\) points, one on \(13\) points, two on \(14\) points, and five on \(15\) points. The proof makes extensive use of the notion of the weight of a point in a \(2\)-coloured finite linear space, as well as the distinction between minimal and non-minimal \(2\)-coloured finite linear spaces. We then use this classification to draw some conclusions on two open problems on the \(2\)-colouring of configurations of points.
Suppose \(G\) is a finite plane graph with vertex set \(V(G)\), edge set \(E(G)\), and face set \(F(G)\). The paper deals with the problem of labeling the vertices, edges, and faces of a plane graph \(G\) in such a way that the label of a face and labels of vertices and edges surrounding that face add up to a weight of that face. A labeling of a plane graph \(G\) is called \(d\)-antimagic if for every number \(s\), the \(s\)-sided face weights form an arithmetic progression of difference \(d\). In this paper, we investigate the existence of \(d\)-antimagic labelings for a special class of plane graphs.
The choice number of a graph \(G\), denoted by \(\chi_l(G)\), is the minimum number \(\chi_l\) such that if we give lists of \(\chi_l\) colors to each vertex of \(G\), there is a vertex coloring of \(G\) where each vertex receives a color from its own list no matter what the lists are. In this paper, we show that \(\chi_l(G) \leq 3\) for each plane graph of girth at least \(4\) which contains no \(8\)-circuits and \(9\)-circuits.
It is noted that Teirlinck’s “transposition argument” for disjoint \(\text{STS}(v)\) applies more generally to certain partial triple systems of different orders. A corollary on the number of blocks common to two \(\text{STS}(v)\) of different orders is also given.
We introduce a generalisation of the traditional magic square, which proves useful in the construction of magic labelings of graphs. An order \(n\) sparse semi-magic square is an \(n \times n\) array containing the entries \(1, 2, \ldots, m\) (for some \(m < n^2\)) once each with the remainder of its entries \(0\), and its rows and columns have a constant sum \(k\). We discover some of the basic properties of such arrays and provide constructions for squares of all orders \(n \geq 3\). We also show how these arrays can be used to produce vertex-magic labelings for certain families of graphs.
A graph \(G\) on \(n\) vertices has a prime labeling if its vertices can be assigned the distinct labels \(1, 2, \ldots, n\) such that for every edge \(xy\) in \(G\), the labels of \(x\) and \(y\) are relatively prime. In this paper, we show that generalized books and \(C_m\) snakes all have prime labelings. In the process, we demonstrate a way to build new prime graphs from old ones.
In this paper, we studied that a linear space, which is the complement of a linear space having points are not on a trilateral or a quadrilateral in a projective subplane of order \(m\), is embeddable in a unique way in a projective plane of order \(n\). In addition, we showed that this linear space is the complement of certain regular hyperbolic plane in the sense of Graves \([5]\) with respect to a finite projective plane.
We give a combinatorial proof of Wilson’s Theorem: \(p\) divides \(\{(p – 1)! +1\}\) if \(p\) is prime.
The Padmakar-Ivan (PI) index of a graph \(G\) is defined as \(PI(G) = \sum[n_{eu} (e|G) + n_{ev}(e|G)]\) where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\) than to \(v\), \(n_{ev}(e|G)\) is the number of edges of \(G\) lying closer to \(v\) than to \(u\), and the summation goes over all edges of \(G\). The PI Index is a Szeged-like topological index developed very recently. In this paper, an exact expression for the PI index of the armchair polyhex nanotubes is given.
A finite planar set is \(k\)-isosceles for \(k \geq 3\), if every \(k\)-point subset of the set contains a point equidistant from the other two. This paper gives a \(4\)-isosceles set consisting of \(7\) points with no three on a line and no four on a circle.
For a group \(T\) and a subset \(S\) of \(T\), the bi-Cayley graph \(\text{BCay}(T, S)\) of \(T\) with respect to \(S\) is the bipartite graph with vertex set \(T \times \{0, 1\}\) and edge set \(\{\{(g, 0), (ag, 1)\} | g \in T, s \in S\}\). In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infinite families of cubic semisymmetric graphs.
We study the notion of path-congruence \(\Phi: T_1 \rightarrow T_2\) between two trees \(T_1\) and \(T_2\). We introduce the concept of the trunk of a tree, and prove that, for any tree \(T\), the trunk and the periphery of \(T\) are stable. We then give conditions for which the center of \(T\) is stable. One such condition is that the central vertices have degree \(2\). Also, the center is stable when the diameter of \(T\) is less than \(8\).
We call a cycle whose length is at most \(5\) a short cycle. In this paper, we consider the packing of short cycles in a graph with specified edges. A minimum degree condition is obtained, which is slightly weaker than that of the result in \([1]\).
Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \(f\)-factor if \(d_G^{h}(x) = f(x)\) for every \(x \in V(F)\). In this paper, we prove that if \(\delta(G) \geq b\) and \(\alpha(G) \leq \frac{4a(\delta-b)}{(b+1)^2}\), then \(G\) has a fractional \(f\)-factor. Where \(a\) and \(b\) are integers such that \(0 \leq a \leq f(x) \leq b\) for every \(x \in V(G)\). Therefore, we prove that the fractional analogue of Conjecture in \([2]\) is true.
Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group, \(g \in A\) and \(\Gamma\) a group of automorphisms of \(D\). We consider the number of \(T\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. Specifically, we enumerate the number of \(I\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order and the trivial automorphism group \(\Gamma\) of \(D\), when \(A\) is the cyclic group \({Z}_{p^n}\) and the direct sum of \(m\) copies of \({Z}_p\) for any prime number \(p (> 2)\).
The Grundy number of an impartial game \(G\) is the size of the unique Nim heap equal to \(G\). We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these “fractal sequences” and give a bijection between these sequences and certain upper-triangular arrays. As a special case, we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap.
A graph \(G\) is clique-perfect if the cardinality of a maximum clique-independent set of \(H\) is equal to the cardinality of a minimum clique-transversal of \(H\), for every induced subgraph \(H\) of \(G\). When equality holds for every clique subgraph of \(G\), the graph is \(c\)-clique-perfect. A graph \(G\) is \(K\)-perfect when its clique graph \(K(G)\) is perfect. In this work, relations are described among the classes of perfect, \(K\)-perfect, clique-perfect and \(c\)-clique-perfect graphs. Besides, partial characterizations of \(K\)-perfect graphs using polyhedral theory and clique subgraphs are formulated.
In this note, we investigate arithmetic properties of the Motzkin numbers. We prove that for large \(n\), the product of the first \(n\) Motzkin numbers is divisible by a large prime. The proofs use the Deep Subspace Theorem.
The point-distinguishing chromatic index of a graph \(G\), denoted by \(\chi_o(G)\), is the smallest number of colours in a (not necessarily proper) edge colouring of \(G\) such that any two distinct vertices of \(G\) are distinguished by sets of colours of their adjacent edges. The exact value of \(\chi_o(K_{m,n})\) is found if either \(m \leq 10\) or \(n \geq 8m^2 – 2m + 1\).
Star graphs were introduced by \([1]\) as a competitive model to the \(n\)-cubes. Then hyper-stars were introduced in \([9]\) to be a competitive model to both \(n\)-cubes and star graphs. In this paper, we discuss strong connectivity properties and orientability of the hyper-stars.
In this paper, three methods for constructing larger harmonious graphs from one or a set of harmonious graphs are provided.
The complexity of determining if a Steiner triple system on \(v = 6n + 3\) points contains a parallel class is currently unknown. In this paper, we show that the problem of determining if a partial Steiner triple system on \(v = 6n + 3\) points contains a parallel class is NP-complete. We also consider the problem of determining the chromatic index of a partial Steiner triple system and show that this problem is NP-hard.
In this paper, it has been proved that \(K_{r,r} \times K_{m}\), \(m \geq 3\), is hamiltonian decomposable.
A twofold extended triple system with two idempotent elements, \(TETS(v)\), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of triples, called blocks, of type \(\{x,y,z\}\), \(\{x,x,y\}\) or \(\{x,x,x\}\) such that every pair of elements of \(V\), not necessarily distinct, belongs to exactly two triples and there are only two triples of the type \(\{x, x, x\}\).
This paper shows that an indecomposable \(TETS(v)\) exists which contains exactly \(k\) pairs of repeated blocks if and only if \(v \not\equiv 0 \mod 3\), \(v \geq 5\) and \(0 \leq k \leq b_v – 2\), where \(b_v = \frac{(v + 2)(v + 1)}{6}\).
For a subset of vertices \(S\) in a graph \(G\), if \(v \in S\) and \(w \in V-S\), then the vertex \(w\) is an \(external\; private\; neighbor\; of \;v\) (with respect to \(S\)) if the only neighbor of \(w\) in \(S\) is \(v\). A dominating set \(S\) is a private dominating set if each \(v \in S\) has an external private neighbor. Bollébas and Cockayne (Graph theoretic parameters concerning domination, independence and irredundance. J. Graph Theory \(3 (1979) 241-250)\) showed that every graph without isolated vertices has a minimum dominating set which is also a private dominating set. We define a graph \(G\) to be a \(private\; domination\; graph\) if every minimum dominating set of \(G\) is a private dominating set. We give a constructive characterization of private domination trees.
The Levi graph of a balanced incomplete block design is the bipartite graph whose vertices are the points and blocks of the design, with each block adjacent to those points it contains. We derive upper and lower bounds on the isoperimetric numbers of such graphs, with particular attention to the special cases of finite projective planes and Hadamard designs.
A strong \( k \)-edge-coloring of a graph \( G \) is an assignment of \( k \) colors to the edges of \( G \) in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of \( G \), are assigned different colors. The strong chromatic index of \( G \) is the smallest number \( k \) for which \( G \) has a strong \( k \)-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree two and a cycle connecting the leaves of the tree. A caterpillar is a tree such that the removal of the leaves becomes a path. In this paper, we show that the strong chromatic index of cubic Halin graph is at most 9. That is, every cubic Halin graph is edge-decomposable into at most 9 induced matchings. Also, we study the strong chromatic index of a cubic Halin graph whose characteristic tree is a caterpillar.
Let \( G \) be a graph of order \( n(G) \), minimum degree \( \delta(G) \), diameter \( d_m(G) \), and let \( \bar{G} \) be the complement of the graph \( G \). A vertex set \( D \) is called a dominating set of \( G \), if each vertex not in \( D \) has at least one neighbor in \( D \). The domination number \( \gamma(G) \) equals the minimum cardinality of a dominating set of \( G \).
In this article we show the inequalities
Using the concept of connectivity, we present some related upper bounds for the domination number of graphs with \( \text{dm}(G) = 2 \) and \( \text{dm}(G) = 3 \).
We prove in this note that certain caterpillars with diameter 4 or 5 do not factorize complete graphs. This together with results by Kovarova [2,3] and Kubesa [5] gives the complete characterization of the caterpillars with diameter 4 that factorize the complete graph \( K_{2n} \). For diameter 5, we again complement results by Kovarova [4] and Kubesa [6-9] to give the complete characterization for certain class of caterpillars.
High-performance computers have been in great demand for applications in different areas. The increase in the processing power of processors cannot solely satisfy our demand. Parallel computers are made to overcome this technology limitation. In the last decade, research topics on parallel computer using network-connected multicomputer have been studied extensively. A cost-efficient high-speed multicomputer system was built using the SCSI bus for the network connection, and it has been shown that it can reduce the communication overheads and hence increase the overall performance [5]. In order to build highly scalable multiple computers based on this design, we have to take into consideration of different network topologies. Since SCSI bus [2,3] possesses some unique properties, it induces some interesting properties on the design of the network topology. In this paper, we evaluate the performance of the large scale SCSI networks with linear and mesh structures.
The degree set of a finite simple graph \( G \) is the set of distinct degrees of vertices of \( G \). For any given finite set \( \mathcal{D} \) of positive integers, we determine all positive integers \( n \) such that \( \mathcal{D} \) is the degree set of some simple graph with \( n \) vertices. This extends a theorem of Kapoor, Polimeni \(\& Wall (1977)\) which shows that the least such \( n \) is \( 1 + \max(\mathcal{D}) \).
Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x,y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). By analogy with graceful labelings, we say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labelled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges every label either occurs exactly \( k \) times or does not occur at all. For \( m \geq 3 \), let \( C_m’ \) (denoted also in the literature by \( C_m \circ K_1 \) and called a corona graph) be a graph with \( 2m \) vertices such that there is a partition of them into sets \( U \) and \( V \) of cardinality \( m \), with the property that \( U \) spans a cycle, \( V \) is independent and the edges joining \( U \) to \( V \) form a matching. Let \( \mathcal{P} \) be the set of all pairs \( (m, k) \) of positive integers such that \( k \) is a proper divisor of \( 2m \) (i.e., a divisor different from \( 2m \) and \( 1 \)) and \( k \) is odd if \( m \) is odd. We show that \( C_m’ \) is minimally \( k \)-equitable if and only if \( (m,k) \in \mathcal{P} \).
We show that the number of points at distance \( i \) from a given point \( x \) in a dense near polygon only depends on \( i \) and not on the point \( x \). We give a number of easy corollaries of this result. Subsequently, we look to the case of dense near polygons \( S \) with an order in which there are two possibilities for \( t_Q \), where \( Q \) is a quad of \( S \), and three possibilities for \( (t_H, v_H) \), where \( H \) is a hex of \( S \). Using the above-mentioned results, we will show that the number of quads of each type through a point is constant. We will also show that the number of hexes of each type through a point is constant if a certain matrix is nonsingular. If each hex is a regular near hexagon, a glued near hexagon or a product near hexagon, then that matrix turns out to be nonsingular in all but one of the eight possible cases. For the exceptional case, however, we provide an example of a near polygon that does not have a constant number of hexes of each type through each point.
In the Euclidean plane, let \( A \), \( B \), \( C \) be noncollinear points and \( T \) be the union of the lines \( AB \), \( BC \), \( CA \). It is shown that there is a point \( P \) such that if \( \tilde{T} \) is the image of \( T \) by any nonrotating uniform expansion about \( P \), then \( T \cap \tilde{T} \) is generally a six-point set that lies on a circle.
We show that for each positive integer \( t \), for which there is a skew-type Hadamard matrix of order \( 4t \), there is a quasi-symmetric \( ((4t – 1)^2, (4t – 1)(2t – 1), t(4t – 3)) \) design.
The Moore upper bound for the order \( n(\Delta, 2) \) of graphs with maximum degree \( \Delta \) and diameter two is \( n(\Delta, 2) < \Delta^2 + 1 \). The only general lower bound for vertex symmetric graphs is \( n_{vt}(\Delta, 2) \geq \left\lfloor \frac{\Delta + 2}{2} \right\rfloor \left\lceil \frac{\Delta + 2}{2} \right\rceil \). Recently, a construction of vertex transitive graphs of diameter two, based on voltage graphs, with order \( \frac{8}{9} \left( \Delta + \frac{1}{2} \right)^2 \) has been given in [5] for \( \Delta = \frac{3q – 1}{2} \) and \( q \) a prime power congruent with 1 mod 4. We give an alternative geometric construction which provides vertex transitive graphs with the same parameters and, when \( q \) is a prime power not congruent to 1 modulo 4, it gives vertex transitive graphs of diameter two and order \( \frac{1}{2} (\Delta + 1)^2 \), where \( \Delta = 2q – 1 \). For \( q = 4 \), we obtain a vertex transitive graph of degree 6 and order 32.
We present an optimal algorithm to label the edges of a complete graph with integer lengths so that every Hamilton cycle has the same length. The algorithm is complete in the sense that every edge-labelling with this property is the output labelling of some run of this algorithm. Such edge-labellings are induced by half-integer vertex-labellings by adding the vertex labels on an edge’s ends to determine its label. The Fibonacci sequence arises in this connection.
Two players are presented with a finite, simple graph \( G = (V, E) \) that has no isolated vertices. They take turns deleting an edge from the graph in such a way that no isolated vertex is created. The winner is the last player able to remove an edge. We analyze this game when the graph \(G\) is a path of arbitrary length. In addition, some observations are made in the situation that the graph has an automorphism of a special type.
A (previously reported) surprising and attractive hypergeometric identity is established from first principles using three hypergeometric transformations.
Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of computing a kind of particular basis of an ideal which enjoys some important properties. In this paper, we introduce the concept of Hadamard ideals in order to establish a new approach to the construction of Hadamard matrices with circulant core. Hadamard ideals reveal the rich interplay between Hadamard matrices with circulant core and ideals in multivariate polynomial rings. Hadamard ideals yield an exhaustive search for Hadamard matrices with circulant core for any specific dimension. In particular, we furnish all solutions for Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core. We establish the dihedral structure of the varieties associated with Hadamard ideals. Finally, we furnish the complete lists (exhaustive search) of inequivalent Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core.
Let \( K_v \) be the complete graph on \( v \) vertices, and \( C_5 \) be a cycle of length five. A simple minimum \( (v, C_5, 1) \)-covering is a pair \( (V, C) \) where \( V = V(K_v) \) and \( C \) is a family of edge-disjoint 5-cycles of minimum cardinality which partition \( E(K_v) \cup E \), for some \( E \subset E(K_v) \). The collection of edges \( E \) is called the excess. In this paper, we determine the necessary and sufficient conditions for the existence of a simple minimum \( (v, C_5, 1) \)-covering. More precisely, for each \( v \geq 6 \), we prove that there is a simple minimum \( (v, C_5, 1) \)-covering having all possible excesses.
The resolution of workshop problems, such as the Flow Shop or the Job Shop, has great importance in industrial areas. Criteria to optimize are generally the minimization of the makespan time or the tardiness time. However, few resolution approaches take into account those different criteria simultaneously. This paper presents a comparative and progressive study of different multicriteria optimization techniques. Several strategies of selection, diversity maintaining, and hybridization will be exposed. Their performances will be compared and tested. A parallel GA model is proposed, which allows increasing the population size and the limit generations number, and leads to better results. In parallel to the work on the optimization technique, we propose here a new bi-criteria flow shop benchmark, responding to the need for common problem instances in the field of multicriteria optimization.
In this paper we determine a class of critical sets in the abelian \(2\)-group that may be obtained from a greedy algorithm. These new critical sets are all \(2\)-critical (each entry intersects an intercalate, a trade of size \(4\)) and complete in a top-down manner.
In a \((k, n)\)-threshold scheme, a secret key \(K\) is split into \(n\) shares in such a way that \(K\) can be recovered from \(k\) or more shares, but no information about \(K\) can be obtained from any \(k-1\) or fewer shares. We are interested in the situation where there are some number of incorrect (i.e., faulty) shares. When there are faulty shares, we might need to examine more than \(k\) shares in order to reconstruct the secret correctly. Given an upper bound, namely \(t\), on the number of faulty shares, we focus on finding efficient algorithms for reconstructing the secret in a \((k, n)\)-threshold scheme. We call this the threshold scheme with cheaters problem.
We first review known combinatorial algorithms that use covering designs, as presented in Rees et al. [11] and Tso et al. [13]. Then we extend the ideas of their algorithms to a more general one. We also link the threshold scheme with cheaters problem to decoding generalized Reed-Solomon codes. Then we adapt two decoding algorithms, namely, the Peterson-Gorenstein-Zierler Algorithm and Gao’s Algorithm, to solve our problem. Finally, we contribute a general algorithm that combines both the combinatorial and decoding approaches, followed by an experimental analysis of all the algorithms we describe.
Let \( G \) be a simple graph, and let \( p \) be a positive integer. A subset \( D \subseteq V(G) \) is a \( p \)-\({dominating}\) set of the graph \( G \), if every vertex \( v \in V(G) – D \) is adjacent to at least \( p \) vertices of \( D \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality among the \( p \)-dominating sets of \( G \). Note that the \( 1 \)-domination number \( \gamma_1(G) \) is the usual domination number \( \gamma(G) \). The covering number of a graph \( G \) is denoted by \( \beta(G) \). If \( T \) is a tree of order \( n(T) \), then Fink and Jacobson [1] proved in 1985 that
\[\gamma_p(T) \geq \frac{(p-1)n(T) + 1}{p}\]
The special case \( p = 2 \) of this inequality easily leads to
\[\gamma_2(T) \geq \beta(T) + 1 \geq \gamma(T) + 1\]
for every non-trivial tree \( T \). Inspired by the article of Fink and Jacobson [1], we characterize in this paper the family of trees \( T \) with \( \gamma_p(T) = \left\lceil \frac{(p-1)n(T) + 1}{p} \right\rceil \) as well as all non-trivial trees \( T \) with \( \gamma_2(T) = \gamma(T) + 1 \) and \( \gamma_2(T) = \beta(T) + 1 \).
Alliances in undirected graphs were introduced by Hedetniemi, Hedetniemi, and Kristiansen, and generalized to \( k \)-alliances by Shafique and Dutton. We translate these definitions of alliances to directed graphs. We establish basic properties of alliances and examine bounds on the size of minimal alliances in directed graphs. In general, the bounds established for alliances in undirected graphs do not hold when alliances are considered over the larger class of directed graphs and we construct examples which break these bounds.
For given integers \( k \) and \( \ell \), \( 3 \leq k \leq \ell \), a graphic sequence \( \pi = (d_1, d_2, \dots, d_n) \) is said to be potentially \({}_{k}C_\ell\)-graphic if there exists a realization of \( \pi \) containing \( C_r \), for each \( r \), where \( k \leq r \leq \ell \) and \( C_r \) is the cycle of length \( r \). Luo (Ars Combinatoria 64(2002)301-318) characterized the potentially \( C_\ell \)-graphic sequences without zero terms for \( r = 3, 4, 5 \). In this paper, we characterize the potentially \({}_{k}C_\ell\)-graphic sequences without zero terms for \( k = 3, 4 \leq \ell \leq 5 \) and \( k = 4, \ell = 5 \).
We show that deciding if a set of vertices is an eternal \(1\)-secure set is complete for \(\text{co-}NP^{\text{NP}}\), solving a problem stated by Goddard, Hedetniemi, and Hedetniemi \([JCMCC, \text{vol. 52}, \text{pp. 160-180}]\).
A Sarvate-Beam type of triple system is defined in the case \( v \equiv 2 \pmod{3} \) and an enumeration is given of such systems for \( v = 5 \).
Informally, a set of guards positioned on the vertices of a graph \( G \) is called eternally secure if the guards are able to respond to vertex attacks by moving a single guard along a single edge after each attack regardless of how many attacks are made. The smallest number of guards required to achieve eternal security is the eternal security number of \( G \), denoted \( es(G) \), and it is known to be no more than \( \theta_v(G) \), the vertex clique cover number of \( G \). We investigate conditions under which \( es(G) = \theta_v(G) \).
We apply Computational Algebra methods to the construction of Hadamard matrices from two circulant submatrices, given by C. H. Yang. We associate Hadamard ideals to this construction, to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search for Hadamard matrices from two circulant submatrices for this construction, for the first eight admissible values \(2, 4, 8, 10, 16, 18, 20, 26\) and partial searches for the next three admissible values \(32, 34, 40\). From the solutions we found, for the admissible values \(26\) and \(34\), we located new inequivalent Hadamard matrices of orders \(52\) and \(68\) with two circulant submatrices, thus improving the lower bounds for the numbers of inequivalent Hadamard matrices of orders \(52\) and \(68\). We also propose a heuristic decoupling of one of the equations arising from this construction, which can be used together with the PSD test to search for solutions more efficiently.
A Hamilton cycle in an \( n \)-cube is said to be \( k \)-warped if its \( k \)-paths have their edges running along different parallel \( 1 \)-factors. No Hamilton cycle in the \( n \)-cube can be \( n \)-warped. The equivalence classes of Hamilton cycles in the \( 5 \)-cube are represented by the circuits associated to their corresponding minimum change-number sequences, or minimum \( H \)-circuits. This makes feasible an exhaustive search of such Hamilton cycles allowing their classification according to class cardinalities, distribution of change numbers, duplicity, reversibility, and \( k \)-warped representability, for different values of \( k < n \). This classification boils down to a detailed enumeration of a total of \( 237675 \) equivalence classes of Hamilton cycles in the \( 5 \)-cube, exactly four of which do not traverse any sub-cube. One of these four classes is the unique class of \( 4 \)-warped Hamilton cycles in the \( 5 \)-cube. In contrast, there is no \( 5 \)-warped Hamilton cycle in the \( 6 \)-cube. On the other hand, there is exactly one class of Hamilton cycles in the graph of middle levels of the \( 5 \)-cube. A representative of this class possesses an elegant geometrical and symmetrical disposition inside the \( 5 \)-cube.
The main objective of this paper is to introduce a generalization of distance called superior distance in graphs. For two vertices \( u \) and \( v \) of a connected graph, we define \( \text{D}_{u,v} = \text{N}[u] \cup \text{N}[v] \). We define a \( \text{D}_{u,v} \)-walk as a \( u \)-\( v \) walk that contains every vertex of \( \text{D}_{u,v} \). The superior distance \( \text{d}_D(u,v) \) from \( u \) to \( v \) is the length of a shortest \( \text{D}_{u,v} \)-walk. In this paper, first we give the bounds for the superior diameter of a graph and a property that relates the superior eccentricities of adjacent vertices. Finally, we investigate those graphs that are isomorphic to the superior center of some connected graph and those graphs that are isomorphic to the superior periphery of some connected graph.
For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \) defined by
\[l^+(v) = \sum_{uv \in E(G)} l(uv)\]
is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). The concept of integer-magic spectrum of a graph was first introduced in [4]. But unfortunately, this paper has a number of incorrect statements and theorems. In this paper, first we will correct some of those statements, then we will determine the integer-magic spectra of caterpillars.
A sequence \( S \) is potentially \( K_{m}-C_4 \)-graphical if it has a realization containing a \( K_m-C_4 \) as a subgraph. Let \( \sigma(K_m-C_4,n) \) denote the smallest degree sum such that every \( n \)-term graphical sequence \( S \) with \( \sigma(S) \geq \sigma(K_m-C_4,n) \) is potentially \( K_m-C_4 \)-graphical. In this paper, we prove that \( \sigma(K_m-C_4,n) \geq (2m-6)n-(m-3)(m-2)+2 \), for \( n \geq m \geq 4 \). We conjecture that equality holds for \( n \geq m \geq 4 \). We prove that this conjecture is true for \( m = 5 \).
In 1975, Leech introduced the problem of finding trees whose edges can be labeled with positive integers in such a way that the set of distances (sums of weights) between vertices is \(\{1, 2, \dots, \binom{n}{2}\}\), where \(n\) is the number of vertices. We refer to such trees as perfect distance trees. More generally, we define a distinct distance tree to be a weighted tree in which the distances between vertices are distinct. In this article, we focus on identifying minimal distinct distance trees. These are the distinct distance trees on \(n\) vertices that minimize the maximum distance between vertices. We determine \(M(n)\), the maximum distance in a minimal distinct distance tree on \(n\) vertices, for \(n \leq 10\), and give bounds on \(M(n)\) for \(n \geq 11\). This includes a determination of all perfect distance trees for \(n < 18\). We then consider trees according to their diameter and show that there are no further perfect distance trees with diameter at most \(3\). Finally, generalizations to graphs, forests, and distinct distance sets are considered.
A bijection \( \lambda: V \cup E \cup F \to \{1, 2, 3, \dots, |V| + |E| + |F|\} \) is called a \( d \)-antimagic labeling of type \( (1, 1, 1) \) of plane graph \( G(V, E, F) \) if the set of \( s \)-sided face weights is \( W_s = \{a_s + a_s+d, a_s+2d, \dots, a_s + (f_s-1)d\} \) for some integers \( s \), \( a_s \), and \( d \), where \( f_s \) is the number of \( s \)-sided faces and the face weight is the sum of the labels carried by that face and the edges and vertices surrounding it. In this paper, we examine the existence of \( d \)-antimagic labelings of type \( (1, 1, 1) \) for a special class of plane graphs \( {C}_a^b \).
GWhD(\(v\))s, or Generalized Whist Tournament Designs on \( v \) players, are a relatively new type of design. GWhD(\(v\))s are (near) resolvable (\(v,k,k-1\)) BIBDs. For \( k = et \), each block of the design is considered to be a game involving \( e \) teams of \( t \) players each. The design is subject to the requirements that every pair of players appears together in the same game exactly \( t-1 \) times as teammates and exactly \( k-t \) times as opponents. These conditions are referred to as the Generalized Whist Conditions, and when met, we refer to the (N)RBIBD as a (\( t, k \)) GWhD(\(v\)). When \( k = 10 \), necessary conditions on \( v \) are that \( v \equiv 0, 1 \pmod{10} \). In this study, we focus on the existence of (\(2,10\)) GWhD(\(v\)), \(v \equiv 1 \pmod{10}\). It is known that a (\(2,10,9\))-NRBIBD does not exist. Therefore, it is impossible to have a (\(2,10\)) GWhD(\(21\)). It is established here that (\(2,10\)) GWhD(\(10n+1\)) exist for all other \(v\) with at most 42 additional possible exceptions.
We define an overfull set of one-factors of \( K_{2n} \) to be a set of one-factors that between them cover all the edges of \( K_{2n} \), but contain no one-factorization of \( K_{2n} \). We address the question: how many members can such a set contain?
This note proves that, given one member, \(T\), of a particular family of radius-three trees, every radius-two, triangle-free graph, \(G\), with large enough chromatic number contains an induced copy of \(T\).
Let \(k(D)\) be the index of convergence of a digraph \(D\) of order \(n \geq 8\). It is proved that if \(D\) is not strong with only minimally strong components and the greatest common divisor of the cycle lengths of \(D\) is at least two, then
\[k(D) \leq \begin{cases}
\frac{1}{2}(n^2 – 8n + 24) & \text{if } n \text{ is even}, \\
\frac{1}{2}(n^2 – 10n + 35) & \text{if } n \text{ is odd}.
\end{cases}\]
The cases of equality are also characterized.
Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that the graphs \(C_n^{(t)}\) are graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 4k\). In this paper, the conjecture is shown to be true for \(n = 7\).
It is well known that a linear code over a finite field with the systematic generator matrix \([I | P]\) is MDS (Maximum Distance Separable) if and only if every square submatrix of \(P\) is nonsingular. In this correspondence, we obtain a similar characterization for the class of Near-MDS codes in terms of the submatrices of \(P\).
In this paper, we define the signed total domatic number of a graph in an analogous way to that of the fractional domatic number defined by Rall (A fractional version of domatic number. Congr. Numer. \(74 (1990), 100-106)\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A set \(\{f_1,\ldots,f_a\}\) of signed total dominating functions on \(G\) such that \(\sum\limits_{i=1}^a f_i(v) \leq 1\) for each vertex \(v \in V(G)\) is called a signed total dominating family of functions on \(G\). The signed total domatic number of \(G\) is the maximum number of functions in a signed total dominating family of \(G\). In this paper, we investigate the signed total domatic number for special classes of graphs.
Let \(G\) be the product of two directed cycles, let \(Z_a\) be a subgroup of \(Z_a\), and let \(Z_d\) be a subgroup of \(Z_b\). Also, let \(A = \frac{a}{c}\) and \(B = \frac{b}{d}\). We say that \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if there is a spanning connected subgraph of \(G\) that has degree \((2, 2)\) at the vertices of \(Z_c \times Z_d\) and degree \((1, 1)\) everywhere else. We show that the graph \(G\) is \((Z_c \times Z_d)\)-hyperhamiltonian if and only if there exist positive integers \(m\) and \(n\) such that \(Am + Bn = AB + 1\), \(gcd(m, n) = 1\) or \(2\), and when \(gcd(m, n) = 2\), then \(gcd(dm, cn) = 2\).
In this paper, it is shown that a partial edge-disjoint decomposition of \(K_{n}\) into kites (that is, into copies of \(K_3\) with a pendant edge attached) can be embedded in a complete edge-disjoint decomposition of \(K_{4t+9}\) into kites for all even \(t \geq 2n\). The proof requires first proving another interesting result, a generalization of an embedding result on symmetric latin squares by L. D. Andersen, following a result by A. Cruse.
Let \(T_n\) be the complete binary tree of height \(n\) considered as the Hasse-diagram of a poset with its root \(1_n\) as the maximum element. For a tree or forest \(T\), we count the embeddings of \(T\) into \(T_n\) as posets by the functions \(A(n;T) = |\{S \subseteq T_n : 1_n \in S, S \cong T\}|\), and \(B(n;T) = |\{S \subseteq T_n : 1_n \notin S, S \cong T\}|\). Here we summarize what we know about the ratio \(A(n;T)/B(n;T)\), in case of \(T\) being a chain or an antichain.
In this paper, the concept of clique number of uniform hypergraph is defined and its relationship with circular chromatic number and clique number is studied. For every positive integer \(k, p\) and \(q\), \(2q \leq p\) we construct a \(k\)-uniform hypergraph with small clique number whose circular chromatic number is equal to \(\frac{p}{q}\). We define the concept and study the properties of \(c\)-perfect \(k\)-uniform hypergraphs.
Given a connected graph \(G\) and a subset \(S\) of vertices, the Steiner distance of \(S\) in \(G\) is the minimum number of edges in a tree in \(G\) that contains all of \(S\). Given a positive integer \(m\), let \(\mu_m(G)\) denote the average Steiner distance over all sets \(S\) of \(m\) vertices in \(G\). In particular, \(\mu_2(G)\) is just the average distance of \(G\), often denoted by \(\mu(G)\). Dankelmann, Oellermann, and Swart \([1]\) conjectured that if \(G\) is a connected graph of order \(n\) and \(3 \leq m \leq n\), then \(\frac{\mu_m(G)}{\mu(G)} \geq 3(\frac{m-1}{m+1})\). In this note, we disprove their conjecture by showing that
\[\lim_{m \to \infty} inf \{ \frac{\mu_m(G)}{\mu(G)} :G \text{ is connected and $n(G)\geq m$} \} = 2.\]
Given a simple graph \(G\) on \(n\) vertices, let \(\sigma_2(G)\) be the minimum sum of the degrees of any two non-adjacent vertices. The graph \(G\) is said to be connected if any two distinct vertices may be joined by a path. It is easy to see that if \(\sigma_2(G) \geq n-1\) then \(G\) is not only connected, but we can choose the connecting path to be of size at most two. Ore \([4]\) proved that if \(\sigma_2(G) \geq n+1\) we may always choose this path to cover all the vertices of \(G\). In this paper we extend these results to systems of vertex disjoint paths connecting two vertex \(k\)-sets of \(G\).
A graph with vertex set \(V\) is said to have a prime labeling if its vertices are labelled with distinct integers from \(\{1, 2, \ldots, |V|\}\) such that for each edge \(xy\), the labels assigned to \(x\) and \(y\) are relatively prime. A graph that admits a prime labeling is called a prime graph. It has been conjectured \([1]\) that when \(n\) is a prime integer and \(m < n\), the planar grid \(P_m \times P_n\) is prime. We prove the conjecture and also that \(P_n \times P_n\) is prime when \(n\) is a prime integer.
The exact values of eleven Ramsey numbers \(r(K_{l_1,n_1} , K_{l_2, n_2})\) where \(3 \leq l_1+n_1, l_2+n_2 \leq 7\) are determined, almost completing the table of all \(66\) such numbers.
In \([1]\) and \([4]\), the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this lemma by applying Burnside’s theorem to the corresponding action by an arbitrary finite cyclic group. Although this idea is not new, by revisiting the constructions in \([1]\) and \([4]\) we derive three divisibility theorems for which the aforementioned classical theorems are, respectively, the cases of a prime divisor, and two of these generalizations are new. Throughout, \(n\) and \(p\) denote positive integers with \(p\) prime and \(\mathbb{Z}_n\) denotes the cyclic group of integers under addition modulo \(n\).
Working on the problem of finding the numbers of lattice points inside convex lattice polygons, Rabinowitz has made several conjectures dealing with convex lattice nonagons and decagons. An intensive computer search preceded a formulation of the conjectures. The main purpose of this paper is to prove some of Rabinowitz’s conjectures. Moreover, we obtain an improvement of a conjectured result and give short proofs of two known results.
Let \(k \geq 1\) be an integer and let \(G = (V, E)\) be a graph. A set \(S\) of vertices of \(G\) is \(k\)-independent if the distance between any two vertices of \(S\) is at least \(k+1\). We denote by \(\rho_k(G)\) the maximum cardinality among all \(k\)-independent sets of \(G\). Number \(\rho_k(G)\) is called the \(k\)-packing number of \(G\). Furthermore, \(S\) is defined to be \(k\)-dominating set in \(G\) if every vertex in \(V(G) – S\) is at distance at most \(k\) from some vertex in \(S\). A set \(S\) is \(k\)-independent dominating if it is both \(k\)-independent and \(k\)-dominating. The \(k\)-independent dominating number, \(i_k(G)\), is the minimum cardinality among all \(k\)-independent dominating sets of \(G\). We find the values \(i_k(G)\) and \(\rho_k(G)\) for iterated line graphs.
The maximum genus, a topological invariant of graphs, was inaugurated by Nordhaus \(et\; al\). \([16]\). In this paper, the relations between the maximum non-adjacent edge set and the upper embeddability of a graph are discussed, and the lower bounds on maximum genus of a graph in terms of its girth and maximum non-adjacent edge set are given. Furthermore, these bounds are shown to be best possible. Thus, some new results on the upper embeddability and the lower bounds on the maximum genus of graphs are given.
The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see SIAM J. Discrete Math. \(15(4) (2002), 519-529)\). A set \(S\) of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set \(S\) (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. We investigate the power domination number of a block graph.
A \((p,q)\)-graph \(G\) in which the edges are labeled \(1,2,3,\ldots,q\) so that the vertex sums are constant, is called supermagic. If the vertex sum mod \(p\) is a constant, then \(G\) is called edge-magic. We investigate the supermagic characteristic of a simple graph \(G\), and its edge-splitting extension \(SPE(G,f)\). The construction provides an abundance of new supermagic multigraphs.
The basis number of a graph \(G\) is defined to be the least integer \(k\) such that \(G\) has a \(k\)-fold basis for its cycle space. We investigate the basis number of the composition of theta graphs, a theta graph and a path, a theta graph and a cycle, a path and a theta graph, and a cycle and a theta graph.
We introduce certain types of surfaces \(M_j^n\), for \(j = 1,2,\ldots,11\) and determine their genus distributions. At the basis of joint trees introduced by Liu, we develop the surface sorting method to calculate the embedding distribution by genus.
Network reliability is an important issue in the area of distributed computing. Most of the early work in this area takes a probabilistic approach to the problem. However, sometimes it is important to incorporate subjective reliability estimates into the measure. To serve this goal, we propose the use of the weighted integrity, a measure of graph vulnerability. The weighted integrity problem is known to be NP-complete for most of the common network topologies including tree, mesh, hypercube, etc. It is known to be NP-complete even for most perfect graphs, including comparability graphs and chordal graphs. However, the computational complexity of the problem is not known for one class of perfect graphs, namely, co-comparability graphs. In this paper, we give a polynomial-time algorithm to compute the weighted integrity of interval graphs, a subclass of co-comparability graphs.
The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. An integer distance graph is a graph \(G(D)\) with the set of all integers as vertex set and two vertices \(u,v \in {Z}\) are adjacent if and only if \(|u-v| \in D\) where the distance set \(D\) is a subset of the positive integers set. Let \(D_{m,k} = \{1,2,\ldots,m\} – \{k\}\) for \(m > k \geq 1\). In this paper, some upper and lower bounds of the vertex linear arboricity of the integer distance graph \(G(D_{m,k})\) are obtained. Moreover, \(vla(G(D_{m,1})) = \lceil \frac{m}{4} \rceil +1\) for \(m \geq 3\), \(vla(G(D_{8l+1,2})) = 2l + 2\) for any positive integer \(l\) and \(vla(G(D_{4q,2})) = q+2\) for any integer \(q \geq 2\).
We determine all spreads of symmetry of the dual polar space \(H^D(2n-1,q^2)\). We use this to show the existence of glued near polygons of type \(H^D(2n_1-1,q^2) \otimes H^D(2n_2-1,q^2)\). We also show that there exists a unique glued near polygon of type \(H^D(2n_1-1,4) \otimes H^D(2n_2-1,4)\) for all \(n_1,n_2 \geq 2\). The unique glued near polygon of type \(H^D(2n-1,4) \otimes Q(2n_2-1,q^2)\) has the property that it contains \(H^D(2n-1,4)\) as a big geodetically closed sub near polygon. We will determine all dense near \((2n+2)\)-gons, \(n \geq 3\), which have \(H^D(2n-1,4)\) as a big geodetically closed sub near polygon. We will prove that such a near polygon is isomorphic to either \(H^D(2n+1,4)\), \(H^D(2n-1,4) \otimes Q(5,2)\) or \(H^D(2n-1,4) \times L\) for some line \(L\) of size at least three.
Given a connected graph \(G\) and two vertices \(u\) and \(v\) in \(G\), \(I_G[u, v]\) denotes the closed interval consisting of \(u\), \(v\) and all vertices lying on some \(u\)–\(v\) geodesic of \(G\). A subset \(S\) of \(V(G)\) is called a geodetic cover of \(G\) if \(I_G[S] = V(G)\), where \(I_G[S] = \cup_{u,v\in S} I_G[u, v]\). A geodetic cover of minimum cardinality is called a geodetic basis. In this paper, we give the geodetic covers and geodetic bases of the composition of a connected graph and a complete graph.
Starlike graphs are the intersection graphs of substars of a star. We describe different characterizations of starlike graphs, including one by forbidden subgraphs. In addition, we present characterizations for a natural subclass of it, the starlike-threshold graphs.
We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, for some of the binary codes obtained from the adjacency matrices of the graphs on \(\binom{n}{3}\) vertices, for \(n \geq 7\), with adjacency defined by the vertices as \(3\)-sets being adjacent if they have zero, one or two elements in common, respectively.
In this paper, we have discussed the dynamic coloring of a kind of planar graph. Let \(G\) be a Pseudo-Halin graph, we prove that the dynamic chromatic number of \(G\) is at most \(4\). Examples are given to show the bounds can be attained.
In this paper, we shall consider acquisition sequences of a graph. The formation of each acquisition sequence is a process that creates an independent set. Each acquisition sequence is a sequence of “acquisitions” which are defined on a graph \( G \) for which each vertex originally has a value of one associated with it. In an acquisition, a vertex transfers all of its value to an adjacent vertex with equal or greater value. For an acquisition sequence, one continues until no more acquisitions are possible. The parameter \( a(G) \) is defined to be the minimum possible number of vertices with a nonzero value at the conclusion of such an acquisition sequence. Clearly, if \( S \) is a set of vertices with nonzero values at the end of some acquisition sequence, then \( S \) is independent, and we call such a set \( S \) an acquisition set. We show that for a given graph \( G \), “Is \( a(G) = 1 \)” is NP-complete, and describe a linear time algorithm to determine the acquisition number of a caterpillar.
The cardinality of the minimal pairwise balanced designs on \( v \) elements with largest block size \( k \) is denoted by \( g^{(k)}(v) \). It is known that \( 31 \leq g^{(4)}(18) \leq 33 \). In this paper, we show that \( g^{(4)}(18) \neq 31 \).
In 1966, Wagner used computational search methods to construct a \([23,14,5]\) code. This code has been examined with much interest since that time, in hopes of finding a geometric construction and possible code extensions. In this article, we give a simple geometric construction for Wagner’s code and consider extensions of this construction.
A graph \( G \) is said to be \( E_k \)-Cordial if there is an edge labeling \( f : E(G) \rightarrow \{0,1,\ldots,k-1\} \) such that, at each vertex \( v \), the sum modulo \( k \) of the labels on the edges incident with \( v \) is \( f(v) \) and it satisfies the inequalities \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \), where \( v_f(s) \) and \( e_f(t) \) are, respectively, the number of vertices labeled with \( s \) and the number of edges labeled with \( t \). The map \( f \) is then called an \( E_k \)-cordial labeling of \( G \).
This paper investigates \( E_3 \)-cordiality of snakes, one point unions, path unions, and coronas involving complete graphs.
A defensive \( k \)-alliance in a graph \( G = (V,E) \) is a set of vertices \( A \subseteq V \) such that for every vertex \( v \in A \), the number of neighbors \( v \) has in \( A \) is at least \( k \) more than the number of neighbors it has in \( V – A \) (where \( k \) is a measure of the strength of the alliance). In this paper, we deal with two types of sets associated with defensive \( k \)-alliances: maximum defensive \( k \)-alliance free and minimum defensive \( k \)-alliance cover sets.
Define a set \( X \subseteq V \) to be maximum defensive \( k \)-alliance free if \( X \) does not contain any defensive \( k \)-alliance and is the largest such set. A set \( Y \subseteq V \) is called a \({minimum\; defensive \; k -alliance \;cover}\) if \( Y \) contains at least one vertex from each defensive \( k \)-alliance and is a set of minimum cardinality satisfying this property. We present bounds on the cardinalities of maximum defensive \( k \)-alliance free and minimum defensive \( k \)-alliance cover sets.
This is the first in a series of three papers in which we investigate a special class of designs that we designate as “Moore-Greig Designs”. The sobriquet is associated with the fact that ideas gleaned from two constructions, one due to E. H. Moore (1896) and the other due to M. Greig (2003), are combined to produce designs that have remarkable properties and features. A Moore-Greig Design is an RBIBD that contains, simultaneously, nested RBIBDs, nested GWhDs, many GWhDs, frames, nested frames, GWhFrames, nested GWhFrames, GWhaFrames, RRDFs, and nested RRDFs. All of these designs are Z-cyclic.
To be more precise, let \( p \) be a prime and let \( \{s_i\}_{i=1}^m \) be a monotone increasing sequence of positive integers such that \( s_i | s_{i+1} \) for all \( i, 1 \leq i \leq m-1 \). Let \( n \) be a positive integer such that \( s_m \leq n \) and \( s_m | n \). A Moore-Greig Design is a Z-cyclic \( (p^n, p^{s_m}, p^{s_m} – 1) \)-RBIBD that contains: (1) a Z-cyclic \( (p^n, p^{s_i}, p^{s_i} – 1) \)-RBIBD for each \( i, 1 \leq i \leq m-1 \), (2) a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhD\( (p^n) \) for each \( i, 1 \leq i \leq m-1 \), (3) for each \( i, 1 \leq i \leq m-1 \), a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhD\(_a(p^n)\) for each \( a = \frac{\alpha}{p^{s_i} – 1} \), \( \alpha = 1, 2, \ldots, p^{s_i} – 2 \), (4) a Z-cyclic \( \{p^{s_m}\} \)-frame of type \( (p^{s_m} – 1)^q \), where \( q = \frac{p^{n} – 1}{p^{s_m} – 1} \), (5) a Z-cyclic \( (p^{s_i}, p^{s_m}) \) GWhFrame of type \( (p^{s_m} – 1)^q \) for each \( i, 1 \leq i \leq m-1 \), (6) a Z-cyclic \( (p^{n} – 1, p^{s_i} – 1, p^{s_i}, 1) \)-RRDF for each \( i, 1 \leq i \leq m \).
Other than a single published example, there is no literature pertaining to GWhDs. Therefore, the infinite classes of GWhDs constructed from the Moore-Greig Designs are the first general results related to this type of design. It is also believed that many of the other designs contained within the infinite classes of Moore-Greig designs are new.
In this paper, Part I, we provide detailed descriptions of both the Moore construction and the Greig construction. In the case of the Moore construction, we supply proofs since such proofs are lacking in Moore’s paper. Also included in this paper is a description of Moore-Greig Designs corresponding to \( m = 2 \) and a discussion is given of the presence of the GWhFrames, nested designs, and the RRDFs. Our methods are such that the constructions are straightforward if one has the (associated) Galois Field.
Anti-Pasch partial Steiner triple systems (anti-Pasch PSTSs) arise in erasure codes, extremal set systems, and combinatorial design theory. Maximal anti-Pasch PSTSs correspond to erasure-resilient codes that are used for handling failures in large disk arrays. These codes support the failure of any set of 3 disks and most sets of 4 disks while having the smallest possible update penalty and check-disk overhead.
In this article, we apply a general algorithm for isomorph-free exhaustive generation of incidence structures to the specific case of anti-Pasch PSTSs. We develop and implement a distributed version of the algorithm, which is experimentally analyzed. Using this implementation, we obtain a complete, isomorph-free catalogue of the maximal anti-Pasch PSTSs of order \( v \), for \( v \leq 15 \). The enumeration and classification results for \( 10 \leq v \leq 14 \) are new and computationally nontrivial.
Consider a lottery scheme consisting of randomly selecting a winning \( t \)-set from a universal \( m \)-set, while a player participates in the scheme by purchasing a playing set of any number of \( n \)-sets from the universal set prior to the draw. The player is awarded a prize if \( k \) or more elements in the winning \( t \)-set match those of at least one of the player’s \( n \)-sets in his playing set (\( 1 \leq k \leq \min\{n,t\} \leq m \)). This is called a \( k \)-prize. The player may wish to design a \emph{smallest} playing set which guarantees the player a \( k \)-prize, no matter which winning \( t \)-set is chosen from the universal set.
In this paper, we consider the optimality of the 302 cardinality 7 (or less) lottery design listings in \verb”BELIC” R: \({Lotto \;Systems\; and\; Toto \;Systems \;to \;win\; Wheel\; Game}\), [online], [cited 2003, October 31], available from: {http://www.xs4all.nl/\(\sim\) rbelic/}, for which \( m > 20 \). It is shown, by means of a computerized search technique, that 192 of these designs are optimal, whilst 78 are not, in which case we provide optimal designs.
Then, an additional 429 upper bounds in the tables of Belic (not necessarily of cardinality 7 or less) are improved; 126 of which are optimal. Thus, apart from the 192 designs that we show to be optimal, 204 new lottery numbers are established in this paper, and a further 304 upper bounds are improved. Finally, the optimality of 54 designs of cardinality 7 or less could not be established; however, in each of these cases, a hitherto best known lower bound is provided.
For a simple graph \( G \), consider an injection \( \mu: V \cup E \rightarrow \mathbb{N} \). If for every vertex \( x \in V \) we have \( \mu(x) + \sum_{y \sim x} \mu(xy) = h \), and for every edge \( xy \in E(G) \) we have \( \mu(x) + \mu(xy) + \mu(y) = k \), for some constants \( h \) and \( k \), then \( \mu \) is a totally magic injection (TMI) of \( G \). Also, \( m_t(G) \) is the smallest number in \( \mathbb{N} \) such that there is a TMI \( \mu: V \cup E \rightarrow \{1, 2, \ldots, m_t(G)\} \). Here we study TMIs and the number \( m_t(G) \) for certain \( G \). One theorem, the Star Theorem, is useful for eliminating many classes of well-known graphs that could have a TMI. For most \( n \) and \( n_j \), the following graphs do not have a TMI: every non-star tree, \( P_n \), \( C_n \), \( W_n \), \( K_n \), and \( K_{n_1, n_2, \ldots, n_p} \). We determine \( m_t(F) \) for every forest \( F \) that has a TMI, and \( m_t(G) \) for every graph \( G \) with \( \leq 6 \) vertices that has a TMI.
For a connected graph \( G \) of order \( n \geq 3 \) and an ordered factorization \( \mathcal{F} = \{G_1, G_2, \ldots, G_k\} \) of \( G \) into \( k \) spanning subgraphs \( G_i \) (\( 1 \leq i \leq k \)), the color code of a vertex \( v \) of \( G \) with respect to \( \mathcal{F} \) is the ordered \( k \)-tuple \( c(v) = (a_1, a_2, \ldots, a_k) \) where \( a_i = \text{deg}_{G_i} v \). If distinct vertices have distinct color codes, then the factorization \( \mathcal{F} \) is called a detectable factorization of \( G \); while the detection number \(\text{det}(G)\) of \( G \) is the minimum positive integer \( k \) for which \( G \) has a detectable factorization into \( k \) factors. We study detectable factorizations of cubic graphs. It is shown that there is a unique graph \( F \) for which the Petersen graph has a detectable \( F \)-factorization into three factors. Furthermore, if \( G \) is a connected cubic graph of order \( \binom{k+2}{3} \) with \( \text{det}(G) = k \), then \( k \equiv 2 \pmod{4} \) or \( k \equiv 3 \pmod{4} \). We investigate the largest order of a connected cubic graph with prescribed detection number.
An \((2,1)\) \(\text{coloring}\) of a graph \( G = (V,E) \) is a vertex coloring \( f : V(G) \rightarrow \{0,1,2,\ldots,k\} \) such that \( |f(u) – f(v)| \geq 2 \) for all \( uv \in E(G) \) and \( |f(u) – f(v)| \geq 1 \) if \( d(u,v) = 2 \). The \({span}\) \( \lambda(G) \) is the smallest \( k \) for which \( G \) has an \( L(2,1) \) \(\text{coloring}\). A \(\text{span coloring}\) is an \( L(2,1) \) coloring whose greatest color is \( \lambda(G) \). An \( L(2,1) \)-\(\text{coloring}\) \( f \) is a full-coloring if \( f : V(G) \rightarrow \{0,1,2,\ldots,\lambda(G)\} \) is onto and \( f \) is an irreducible no-hole coloring (inh-coloring) if \( f : V(G) \rightarrow \{0,1,2,\ldots,k\} \) is onto for some \( k \) and there does not exist an \( L(2,1) \)-\(\text{coloring}\) \( g \) such that \( g(u) < f(u) \) for all \( u \in V(G) \) and \( g(v) < f(v) \) for some \( v \in V(G) \). The Assignment sum of \( f \) on \( G \) is the sum of all the labels assigned to the vertices of \( G \) by the \( L(2,1) \) \(\text{coloring}\) \( f \). The \({Sum \;coloring\; number}\) of \( G \), introduced in this paper, \( \sum(G) \), is the minimum assignment sum over all the possible \( L(2,1) \) colorings of \( G \). \( f \) is a \(\text{Sum coloring}\) on \( G \) if its assignment sum equals the \({Sum\; coloring\; number}\). In this paper, we investigate the \({Sum\; coloring\; numbers}\) of certain classes of graphs. It is shown that \( \sum(P_n) = 2(n – 1) \) and \( \sum(C_n) = 2n \) for all \( n \). We also give an exact value for the Sum coloring number of a star and conjecture a bound for the Sum coloring number of an arbitrary graph \( G \) with span \( \lambda(G) \).
Let \( U(n, f) \) denote the graph with vertex set the set of unlabeled graphs of order \( n \) that have no vertex of degree greater than \( f \). Two vertices \( H \) and \( G \) of \( U(n, f) \) are adjacent if and only if \( H \) and \( G \) differ (up to isomorphism) by exactly one edge. The problem of determining the values of \( n \) and \( f \) for which \( U(n, f) \) contains a Hamilton path is investigated. There are only a few known non-trivial cases for which a Hamilton path exists. Specifically, these are \( U(5, 3) \), \( U(6, 3) \), and \( U(7, 3) \). On the other hand, there are many cases for which it is shown that no Hamilton path exists. The complete solution of this problem is unresolved.
We study nega-cyclic \(\pm 1\) matrices. We obtain preliminary results which are then used to decrease the search space. We find that there are \( 2 \), \( 4 \), \( 9 \), \( 23 \), \( 63 \), and \( 187 \) ip-equivalence classes for lengths \( 3 \), \( 5 \), \( 7 \), \( 9 \), \( 11 \), and \( 13 \) respectively. The matrices we find are used in a variant given here of the Goethals-Seidel array to form Hadamard matrices, the aim being to later check them for suitability for CDMA schemes.
Let \(\gamma(G)\) be the domination number of a graph \(G\). A class \(\mathcal{P}\) of graphs is called \(\gamma\)-complete if the problem of determining \(\gamma(G)\), \(G \in \mathcal{P}\), is NP-complete. A class \(\mathcal{P}\) of graphs is called \(\gamma\)-polynomial if there is a polynomial-time algorithm for calculating \(\gamma(G)\) for all graphs \(G \in \mathcal{P}\).
We denote \(\Gamma = \{P_k\cap nK_1 : k \leq 4 \text{ and } n \geq 0\}\). Korobitsin \([4]\) proved that if \(\mathcal{P}\) is a hereditary class defined by a unique forbidden induced subgraph \(H\), then
We extend a positive result (i) in the following way. The class \(\Gamma\) is hereditary, and it is characterized by the set
\[Z(\Gamma) = \{2K_2, K_{1,3},C_3,C_4,C_5\}\]
of minimal forbidden induced subgraphs.
For each \(Z \subseteq Z(\Gamma)\) we consider a hereditary class \({FIS}(Z)\) defined by the set \(Z\) of minimal forbidden induced subgraphs. We prove that \(\mathcal{FIS}(Z)\) is \(\gamma\)-complete in 16 cases, and it is \(\gamma\)-polynomial in the other 16 cases. We also prove that \(2K_2\)-free graphs with bounded clique number constitute a \(\gamma\)-polynomial class.
Let \(G = (V, E)\) be a connected graph and \(S \subseteq E\). \(S\) is said to be an \(r\)-restricted edge cut if \(G – S\) is disconnected and each component in \(G – S\) contains at least \(r\) vertices. Define \(\lambda^{(r)}(G)\) to be the minimum size of all \(r\)-restricted edge cuts and \(\xi_r(G) = \min\{w(U): U \subseteq V, |U| = r\) and the subgraph of \(G\) induced by \(U\) is connected, where \(w(U)\) denotes the number of edges with one end in \(U\) and the other end in \(V\setminus U\). A graph \(G\) with \(\lambda^{(r)} = \xi_r(G)\) (\(r = 1,2,3\)) is called an \(\lambda^{(3)}\)-optimal graph. In this paper, we show that the only edge-transitive graphs which are not \(\lambda^{(3)}\)-optimal are the star graphs \(K_{1, n-1}\), the cycles \(C_n\), and the cube \(Q_3\). Based on this, we determine the expressions of \(N_i(G)\) (\(i = 0,1,\ldots,\xi_3(G) – 1\)) for edge transitive graph \(G\), where \(N_i(G)\) denotes the number of edge cuts of size \(i\) in \(G\).
A vertex-deleted subgraph (subdigraph) of a graph (digraph) \(G\) is called a card of \(G\). A card of \(G\) with which the degree (degree triple) of the deleted vertex is also given is called a degree associated card or dacard of \(G\). To investigate the failure of digraph reconstruction conjecture and its effect on Ulam’s conjecture, we study the parameter \(\textbf{degree associated reconstruction number}\) \(drm(G)\) of a graph (digraph) \(G\) defined as the minimum number of dacards required in order to uniquely identify \(G\). We find \(drm\) for some classes of graphs and prove that for \(t\geq 2\), \(drm(tG)\leq 1+drm(G)\) when \(G\) is connected nonregular and \(drm(tG)\leq m+2-r\) when \(G\) is connected \(r\)-regular of order \(m>2\) and these bounds are tight. \(drm \leq 3\) for other disconnected graphs. Corresponding results for digraphs are also proved.
Two parameters for measuring irregularity in graphs are the degree variance and the discrepancy. We establish best possible upper bounds for the discrepancy in terms of the order and average degree of the graph, and describe some extremal graphs, thereby providing analogues of results of [1], [4] and [5] for the degree variance.
It is calculated the number of symmetric \(r\)-colorings of vertices of a regular \(n\)-gon and the number of equivalence classes of symmetric \(r\)-colorings. A coloring is symmetric if it is invariant with respect to some mirror symmetry with an axis crossing the center of polygon and one of its vertices. Colorings are equivalent if we can get one from another by rotating about the polygon center.
A graph \(G\) is said to be excellent if, given any vertex \(x\) of \(G\), there is a \(\gamma\)-set of \(G\) containing \(x\). It is known that any non-excellent graph can be imbedded in an excellent graph. For example, for every graph \(G\), its corona \(G \circ K\) is excellent, but the difference \(\gamma(G \circ K) – \gamma(G)\) may be high. In this paper, we give a construction to imbed a non-excellent graph \(G\) in an excellent graph \(H\) such that \(\gamma(H) \leq \gamma(G) + 2\). We also show that, given a non-excellent graph \(G\), there is a subdivision of \(G\) which is excellent. The excellent subdivision number of a graph \(G\), \(ESdn{G}\), is the minimum number of edges of \(G\) to be subdivided to get an excellent subdivision graph \(H\). We obtain upper bounds for \(ESdn{G}\). If any one of these upper bounds for \(ESdn{G}\) is attained, then the set of all vertices of \(G\) which are not in any \(\gamma\)-set of \(G\) is an independent set.
In this paper, using the \(q\)-exponential operator technique to Bailey’s \(\mathop{_2\psi_2}\) transformation, we obtain some interesting \(\mathop{_3\psi_3}\)
transformation formulae and summation theorems.
