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

1. \(n \geq 2d+p+2\) when \(p \leq k-2\),
2. \(n \geq 2\left\lceil \frac{d(p+2)}{k} \right\rceil +p +2\) when \(p \geq k-1\).

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 \(C_n\).

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 \(t + 1\).

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 \(R(p)\).

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.