In this paper, we will be concerned with graphs \(G\) satisfying: \(G\) is isometrically embeddable in a hypercube; \(|C(a,b)| = |C(b,a)|\) for every edge \([a,b]\) of \(G\). where \(C(a,b)\) is the set of vertices nearer to \(a\) than to \(b\). Some properties of such graphs are shown; in particular, it is shown that all such graphs \(G\) are \(3\)-connected if \(G\) has at least two edges and \(G\) is not a cycle.

We improve upon Caro’s general polynomial characterizations, all in terms of modified line graphs, restricted to decomposing a graph into isomorphic subgraphs \(H\) with two edges. Firstly, we solve the problem for a multigraph; secondly, we decrease the polynomial bound on complexity if \(H = 2K_2\) and provide an original sufficient condition which can be verified in linear time if \(H = P_3\).

It has been shown by Sittampalam and Keedwell that weak critical sets exist for certain latin squares of order six and that previously claimed examples (for certain latin squares of order \(12\)) are incorrect. This led to the question raised in the title of this paper. Our purpose is to show that five is the smallest order for which weakly completable critical sets exist. In the process of proving this result, we show that, for each of the two types of latin square of order four, all minimal critical sets are of the same type.

We show that if \(G\) is a \((2k-1)\)-connected graph \((k \geq 2)\) with radius \(r\), then \(r \leq \left\lfloor \frac{|V(G)|+2k+9}{2k}\right\rfloor\).

A Cayley digraph \({Cay}(G, S)\) of a finite group \(G\) is isomorphic to another Cayley digraph \({Cay}(G, T)\) for each automorphism \(\sigma\) of \(G\). We will call \({Cay}(G, S)\) a CI-graph if, for each Cayley digraph \({Cay}(G,T)\), whenever \({Cay}(G, S) \cong {Cay}(G,T)\) there exists an automorphism \(\sigma\) of \(G\) such that \(S^\sigma = T\). Further, for a positive integer \(m\), if all Cayley digraphs of \(G\) of out-valency \(m\) are CI-graphs, then \(G\) is said to have the \(m\)-DCI property. This paper shows that for any positive integer \(m\), if a finite abelian group \(G\) has the \(m\)-DCI property, then all Sylow subgroups of \(G\) are homocyclic.

A directed graph operation called pushing a vertex is studied. When a vertex is pushed, the orientation of each of its incident edges is reversed. We consider the problems of pushing vertices so as to produce: strongly connected digraphs semi-connected digraphs acyclic digraphs NP-completeness results are shown for each problem. It is shown that it is possible to create a directed path between any two vertices in a digraph; additional positive results and characterizations are shown for: tournaments outerplanar digraphs Hamiltonian cycles.

A Freeman-Youden rectangle (FYR) is a Graeco-Latin row-column design consisting of a balanced superimposition of two Youden squares. There are well known infinite series of FYRs of size \(q \times (2q+1)\) and \((q+1) \times (2q+1)\) where \(2q+1\) is a prime power congruent to \(3\) (modulo \(4\)). However, Preece and Cameron [9] additionally gave a single FYR of size \(7 \times 15\). This isolated example is now shown to belong to one of a set of infinite series of FYRs of size \(q \times (2q+1)\) where \(q\), but not necessarily \(2q+1\), is a prime power congruent to \(3\) (modulo \(4\)), \(q > 3\); there are associated series of FYRs of size \((q+1) \times (2q+1)\). Both the old and the new methodologies provide FYRs of sizes \(q \times (2q+1)\) and \((q+1) \times (2q+1)\) where both \(q\) and \(2q+1\) are congruent to \(3\) (modulo \(4\)), \(q > 3\); we give special attention to the smallest such size, namely \(11 \times 23\).

Let \(n_4(k,d)\) and \(d_4(n, k)\) denote the smallest value of \(n\) and the largest value of \(d\), respectively, for which there exists an \([n, k, d]\) code over the Galois field \(GF(4)\). It is known (cf. Boukliev [1] and Table B.2 in Hamada [6]) that (1) \(n_4(5, 179) =240\) or \(249\), \(n_4(5,181) = 243\) or \(244, n_4(5, 182) = 244\) or \(245, n_4(5, 185) = 248\) or \(249\) and (2) \(d_4(240,5) = 178\) or \(179\) and \(d_4(244,5) = 181\) or \(182\). The purpose of this paper is to prove that (1) \(74(5,179) = 241, n_4(5,181) = 244, n_4(5,182) = 245, n_4(5, 185) = 249\) and (2) \(d_4(240, 5) = 178\) and \(d_4(244,5) = 181\).

Let \(T_n\) denote any rooted tree with \(n\) nodes and let \(p = p(T_n)\) and \(q = q(T_n)\) denote the number of nodes at even and odd distance, respectively, from the root. We investigate the limiting distribution, expected value, and variance of the numbers \(D(T_n) = |p – q|\) when the trees \(T_n\) belong to certain simply generated families of trees.

In this paper, magic labelings of graphs are considered. These are labelings of the edges with integers such that the sum of the labels of incident edges is the same for all vertices. We particularly study positive magic labelings, where all labels are positive and different. A decomposition in terms of basis-graphs is described for such labelings. Basis-graphs are studied independently. A characterization of an algorithmic nature is given, leading to an integer linear programming problem. Some relations with other graph theoretical subjects, like vertex cycle covers, are discussed.

A forest in which every component is a path is called a path forest. A family of path forests whose edge sets form a partition of the edge set of a graph \(G\) is called a path decomposition of a graph \(G\). The minimum number of path forests in a path decomposition of a graph \(G\) is the linear arboricity of \(G\) and denoted by \(\ell(G)\). If we restrict the number of edges in each path to be at most \(k\) then we obtain a special decomposition. The minimum number of path forests in this type of decomposition is called the linear \(k\)-arboricity and denoted by \(\ell\alpha_k(G)\). In this paper we concentrate on the special type of path decomposition and we obtain the answers for \(\ell\alpha_2(G)\) when \(G\) is \(K_{n,n}\). We note here that if we restrict the size to be one, the number \(\ell\alpha_1(G)\) is just the chromatic index of \(G\).

Primal graphs and primary graphs are defined and compared. All primary stars, paths, circuits, wheels, theta graphs, caterpillars, and echinoids are found, as are all primary graphs of the form \(K_{2,n}\) with \(n \leq 927\).

Let \(K(n | t)\) denote the complete multigraph containing \(n\) vertices and exactly \(t\) edges between every pair of distinct vertices, and let \(f(m,t)\) be the minimum number of complete bipartite subgraphs into which the edges of \(K(n|t)\) can be decomposed. Pritikin [3] proved that \(f(n;t) \geq \max\{n-1,t\}\), and that \(f(m;2) = n\) if \(n = 2,3,5\), and \(f(m;2) = n-1\), otherwise. In this paper, for \(t \geq 3\) using Hadamard designs, skew-Hadamard matrices and symmetric conference matrices [6], we give some complete multigraph families \(K(n|t)\) with \(f(n;t) = n-1\).

Let \(G\) be a graph with \(r \geq 0\) special vertices, \(b_1, \ldots, b_r\), called pins. \(G\) can be composed with another graph \(H\) by identifying each \(b_i\) with another vertex \(a_i\) of \(H\). The resulting graph is denoted \(H \circ G\). Let \(\Pi\) denote a decision problem on graphs. We consider the problem of constructing a “small” minor \(G^*\) of \(G\) that is “equivalent” to \(G\) with respect to the problem \(\Pi\). Specifically, \(G^*\) should satisfy the following:

(C1) \(G^b\) has the same pins as \(G\).

(C2) \(\Pi(H \circ G^b) = \Pi(H \circ G)\) for every \(H\) for which \(H \circ G\) is defined.

(C3) \(|V(G^b)| + |E(G^b)| \leq c \cdot p\), where \(p\) is the number of pins of \(G\), \\and \(c\) is a constant depending only on \(\Pi\).

(C4) \(G^b\) is a minor of \(G\).

We provide linear-time algorithms for constructing such graphs when \(\Pi\) stands for either series-parallelness or outer-planarity. These algorithms lead to linear-time algorithms that determine whether a hierarchical graph is series-parallel or outer-planar and to linear-space algorithms for generating a forbidden subgraph of a hierarchical graph, when one exists.

In this paper, we prove that if \(G\) is a 3-connected planar graph and contains no vertex of degree \(4\), then \(G\) is edge reconstructible. This generalizes a result of J. Lauri [J1].

In this paper, we present a new generalization of the self-complementary graphs, called the \(t\)-sc graphs. Various properties of this class of graphs are studied, generalizing earlier results on self-complementary graphs. Certain existential results on \(t\)-sc graphs are presented, followed by the construction of some infinite classes of \(t\)-sc graphs. Finally, the notion of \(t\)-sc graphs is linked with the notion of factorization, leading to a generalization of \(r\)-partite self-complementary graphs.

In [1], we introduced the generalized exponent for primitive matrices. In this paper, the generalized exponents of tournament matrices are derived.

We provide graceful labelings for prisms \(C_{2m} \times P_n\), with even cycles, for all \(n \geq 2\), and prisms \(C_{2m+1} \times P_n\), with odd cycles when \(3 \leq mn \leq 12\). Further, we verify that the windmill graph \(K^{(m)}_{4}\) is graceful for \(r \leq 22\), and that the square of a path \(P_n\) is graceful for \(n \leq 32\).

In this paper, a composition result, \(viz\)., the number of \(r\)-compositions of \(n\) dominated by the \(r\)-compositions of \(m\) (\(m \geq n\)) subject to certain restrictions, has been derived by the method of induction.

Let \(G\) be a simple graph with \(n\) vertices. Let \(L(G)\) denote the line graph of \(G\). We show that if \(\kappa'(G) > 2\) and if for every pair of nonadjacent vertices \(v,u \in V(G)\), \(d(v) + d(u) > \frac{2n}{3} – 2\), then for any pair of vertices \(e, e’ \in V(L(G))\), either \(L(G)\) has a Hamilton \((e, e’)\)-path, or \(\{e, e’\}\) is a vertex-cut of \(L(G)\). When \(G\) is a triangle-free graph, this bound can be reduced to \(\frac{n}{3}\). These bounds are all best possible and they partially improve prior results in [J.Graph Theory 10(1986),411-425] and [Discrete Math.76(1989)95-116].

The link length of a walk in a multidimensional grid is the number of straight line segments constituting the walk. Alternatively, it is the number of turns that a mobile unit needs to perform in traversing the walk. A rectilinear walk consists of straight line segments which are parallel to the main axis. We wish to construct rectilinear walks with minimal link length traversing grids. If \(G\) denotes the multidimensional grid, let \(s(G)\) be the minimal link length of a rectilinear walk traversing all the vertices of \(G\). In this paper, we develop an asymptotically optimal algorithm for constructing rectilinear walks traversing all the vertices of complete multidimensional grids and analyze the worst-case behavior of \(s(G)\), when \(G\) is a multidimensional grid.

We proved that if a graph \(G\) of minimum valency \(\delta=6\alpha + 5\), with \(\alpha\) a non-negative integer, can triangulate a surface \(\Sigma\) with \(\chi(\Sigma) = -\alpha n + \beta\), where \(\beta \in \{0, 1, 2\}\), then \(G\) is edge reconstructible.

We introduce a concept of “pseudo dual” pseudographs which can be thought of as generalizing some of the recent work on iterated clique graphs. In particular, we characterize those pseudographs which have pseudo duals and show that they encompass several natural families of intersection pseudographs.

Let \(G\) be a simple graph, \(a\) and \(b\) integers and \(f: E(G) \to \{a,a+1,\ldots,b\}\) an integer-valued function with \(\sum_{e\in E(G)} f(e) \equiv 0 \pmod{2}\). We prove the following results:(1) If \(b \geq a \geq 2\), \(G\) is connected and \(\delta(G) \geq \max\left[\frac{b}{2}+2, \frac{(a+b+2)^2}{8a}\right]\), then the line graph \(L(G)\) of \(G\) has an \(f\)-factor;(2) If \(b\geq a \geq 2\), \(G\) is connected and \(\delta(L(G)) \geq \frac{(a+2b+2)^2}{8a}\), then \(L(G)\) has an \(f\)-factor.

We show that a cubic graph is \(\frac{3}{2}\)-tough if and only if it is equal to \(K_4\) or \(K_3 \times K_3\) or else is the inflation of a 3-connected cubic graph.

A directed triple system of order \(v\), denoted \(DT S(v)\), is said to be \(k\)-near-rotational if it admits an automorphism consisting of \(3\) fixed points and \(k\) cycles of length \(\frac{v-3}{3}\). In this paper, we give necessary and sufficient conditions for the existence of \(k\)-near-rotational \(DT S(v)\)s.

A correspondence between decompositions of complete directed graphs with loops into collections of closed trails which partition the edge set of the graph and the variety of all column latin groupoids is established. Subvarieties which consist of column latin groupoids arising from decompositions where only certain trail lengths occur are examined. For all positive integers \(m\), the set of values of \(n\) for which the complete directed graph with loops on a vertex set of cardinality \(n\) can be decomposed in this manner such that all the closed trails have length \(m\) is shown to be the set of all \(n\) for which \(m\) divides \(n^2\).

Let \(X\) be a graph and let \(\alpha(X)\) and \(\alpha'(X)\) denote the domination number and independent domination number of \(X\), respectively. We show that for every triple \((m,k,n)\), \(m \geq 5\), \(2 \leq k \leq m\), \(n > 1\), there exist \(m\)-regular \(k\)-connected graphs \(X\) with \(\alpha'(X) – \alpha(X) > n\). The same also holds for \(m = 4\) and \(k \in \{2,4\}\).

Let \(k\) be an integer greater than one. A set \(S\) of integers is called \(k\)-multiple-free (or \(k\)-MF for short) if \(x \in S\) implies \(kx \notin S\). Let \(f_k(n) = \max\{|A| : A \subseteq [1,n] \text{ is } k\text{-MF}\}\). A subset \(A\) of \([1,n]\) with \(|A| = f_k(n)\) is called a maximal \(k\)-MF subset of \([1,n]\). In this paper, we give a recurrence relation and a formula for \(f_k(n)\). In addition, we give a method for constructing a maximal \(k\)-MF subset of \([1,n]\).

This paper concerns the domination numbers \(\gamma_{k,n}\) for the complete \(k \times n\) grid graphs for \(1 \leq k \leq 10\) and \(n \geq 1\). These numbers were previously established for \(1 \leq k \leq 4\). Here we present dominating sets for \(5 \leq k \leq 10\) and \(n \geq 1\). This gives upper bounds for \(\gamma_{k,n}\) for \(k\) in this range. We discuss evidence that indicates that these upper bounds are also lower bounds.

By a graph we mean an undirected simple graph. The genus \(\gamma(G)\) of a graph \(G\) is the minimum genus of the orientable surface on which \(G\) is embeddable. The thickness \(\Theta(G)\) of \(G\) is the minimum number of planar subgraphs whose union is \(G\).

In [1], it is proved that, if \(\gamma(G) = 1\), then \(\Theta(G) = 2\). If \(\gamma(G) = 2\), the known best upper bound on \(\Theta(G)\) is \(4\) and, as far as the author knows, the known best lower bound is \(2\). In this paper, we prove that, if \(\gamma(G) = 2\), then \(\Theta(G) \leq 3\).

A generalization of (binary) balanced incomplete block designs is to allow a treatment to occur in a block more than once, that is, instead of having blocks of the design as sets, allow multisets as blocks. Such a generalization is referred to as an \(n\)-ary design. There are at least three such generalizations studied in the literature. The present note studies the relationship between these three definitions. We also give some results for the special case when \(n\) is \(3\).

We investigate searching strategies for the set \(\{1, \ldots, n\}\) assuming a fixed bound on the number of erroneous answers and forbidding repetition of questions. This setting models the situation when different processors provide answers to different tests and at most \(k\) processors are faulty. We show for what values of \(k\) the search is feasible and provide optimal testing strategies if at most one unit is faulty.

Ramsey’s Theorem implies that for any graph \(H\) there is a least integer \(r = r(H)\) such that if \(G\) is any graph of order at least \(r\) then either \(G\) or its complement contains \(H\) as a subgraph. For \(n<r\) and \(0 \leq e \leq \frac{1}{2}n(n-1)\), let \(f(e) =1\) {if every graph \(G\) of order \(n\) and size \(e\) is such that either \(G\) or \(\overline{G}\) contains \(H\),} and let \(f(e)=0\) {otherwise.} This associates with the pair \((H,n)\) a binary sequence \(S(H,n)\). By an interval of \(S(H,n)\) we mean a maximal string of equal terms. We show that there exist infinitely many pairs \((H,n)\) for which \(S(H,n)\) has seven intervals.

In this paper the existence of the \(12140\) non-isomorphic symmetric \((49,16,5)\) designs with an involutory homology (this is a special kind of involution acting on a design) is propagated. The automorphism groups are cyclic of orders \(2\), \(4\), \(8\), or \(10\). \(218\) designs are self-dual. The \(40\) designs with an automorphism group of order \(10\) were already given in [2]. A computer (IBM \(3090\)) was used for about \(36\) hours CPU time. According to [2,4] now there are known \(12146\) symmetric \((49,16,5)\) designs.

This paper deals with the joint distributions of some characteristics of the two-dimensional simple symmetric random walk in which a particle at any stage moves one unit in any one of the four directions, namely, north, south, east, and west with equal probability.

In \(1974\), G. Chartrand and R.E. Pippert first defined locally connected and locally \(n\)-connected graphs and obtained some interesting results. In this paper we first extend these concepts to digraphs. We obtain generalizations of some results of Chartrand and Pippert and establish relationships between local connectedness and global connectedness in digraphs.

An infinite countable Steiner triple system is called universal if any countable Steiner triple system can be embedded into it. The main result of this paper is the proof of non-existence of a universal Steiner triple system.

The fact is proven by constructing a family \(\mathcal{S}\) of size \(2^{\omega}\) of infinite countable Steiner triple systems so that no finite Steiner triple system can be embedded into any of the systems from \(\mathcal{S}\) and no infinite countable Steiner triple system can be embedded into any two of the systems from \(\mathcal{S}\) (it follows that the systems from \(\mathcal{S}\) are pairwise non-isomorphic).

A Steiner triple system is called rigid if the only automorphism it admits is the trivial one — the identity. An additional result presented in this paper is a construction of a family of size \(2^{\omega}\) of pairwise non-isomorphic infinite countable rigid Steiner triple systems.

A graph \(G\) having a \(1\)-factor is called \(n\)-extendable if every matching of size \(n\) extends to a 1-factor. We show that
if \(G\) is a connected graph of order \(2p (p \geq 3)\), and g and n are integers, \(1 \leq n < q < p\), such that every induced connected subgraph of order \(2q\) is \(n\)-extendable, then \(G\) is n-extendable.

Certain graphs whose vertices are some collection of subsets of a fixed \(n\)-set, with edges determined by set intersection in some way, have long been conjectured to be Hamiltonian. We are particularly concerned with graphs whose vertex set consists of all subsets of a fixed size \(k\), with edges determined by empty intersection, on the one hand, and with bigraphs whose vertices are all subsets of either size \(k\) or size \(n-k\), with adjacency determined by set inclusion, on the other. In this note, we verify the conjecture for some classes of these graphs. In particular, we show how to derive a Hamiltonian cycle in such a bigraph from a Hamiltonian path in a quotient of a related graph of the first kind (based on empty intersection). We also use a recent generalization of the Chvatal-Erdos theorem to show that certain of these bigraphs are indeed Hamiltonian.

We determine the minimal number of queries sufficient to find an unknown integer \(x\) between \(1\) and \(n\) if at most one answer may be erroneous. The admissible form of query is: “Which one of the disjoint sets \(A_1, \ldots, A_k\) does \(x\) belong to?”

A \(\lambda\)-packing of pairs by quintuples of a \(v\)-set \(V\) is a family of \(5\)-subsets of \(V\) (called blocks) such that every \(2\)-subset of \(V\) occurs in at most \(\lambda\) blocks. The packing number is defined to be the maximum number of blocks in such a \(\lambda\)-packing. These numbers are determined here for \(\lambda \equiv 0 \mod 4\) and all integers \(v \geq 5\) with the exceptions of \((v, \lambda) \in \{(22, 16), (22, 36), (27, 16)\}\).

Recently, there has been substantial interest in the problem of the spectrum of possible support sizes of different families of BIB designs. In this paper, we first prove some theorems concerning the spectrum of any \(t\)-design with \(v = 2k\) and \(k = t + 1\), and then we study the spectrum of the case \(4-(10, 5, 6m)\) in more detail.

We obtain bounds for the separation number of a graph in terms of simpler parameters. With the aid of these bounds, we determine the separation number for various special graphs, in particular multiples of small graphs. This leads to concepts like robustness and asymptotic separation number.

A.M. Assaf, A. Hartman, and N. Shalaby determined in [1] the packing numbers \(\sigma(v, 6, 5)\) for all integers \(v \geq 6\), leaving six open cases of \(v = 41, 47, 53, 59, 62,\) and \(71\). In this paper, we deal with these open cases and thus complete the packing problem.

A hypergraph \(H\) is called connected over a graph \(G\) with the same vertex set as \(H\) if every hyperedge of \(H\) induces a connected subgraph in \(G\). A graph \(F\) is representable in the graph \(G\) if there is some hypergraph \(H\) which is connected over \(G\) and has \(F\) as its intersection graph. Generalizing the well-known problem of representability in forests, the following problems are investigated: Which hypergraphs are connected over some \(n\)-cyclomatic graph, and which graphs are representable in some \(n\)-cyclomatic graph, for any fixed integer \(n\)? Several notions developed in the theory of subtree hypergraphs and chordal graphs (i.e. in the case \(n = 0\)) yield necessary or sufficient conditions, and in certain special cases even characterizations.

Let \(s\) and \(r\) be positive integers with \(s \geq r\) and let \(G\) be a graph. A set \(I\) of vertices of \(G\) is an \((r, s)\)-set if no two vertices of \(I\) are within distance \(r\) from each other and every vertex of \(G\) not in \(I\) is within distance \(s\) from some vertex of \(I\). The minimum cardinality of an \((r, s)\)-set is called the \((r, s)\)-domination number and is denoted by \(i_{r,s}(G)\). It is shown that if \(G\) is a connected graph with at least \(s > r \geq 1\) vertices, then there is a minimum \((r,s)\)-set \(I\) of \(G\) such that for each \(v \in I\), there exists a vertex \(w \in V(G) – I\) at distance at least \(s-r\) from \(v\), but within distance \(s\) from \(v\), and at distance greater than \(s\) from every vertex of \(I – \{v\}\). Using this result, it is shown that if \(G\) is a connected graph with \(p \geq 9 \geq 2\) vertices, then \(i_{r,s}(G) < p/s\) and this bound is best possible. Further, it is shown that for \(s \in \{1,2,3\}\), if \(T\) is a tree on \(p \geq s +1\) vertices, then \(i_{r,s}(T) \leq p/(s +1)\) and this bound is sharp.

We consider the problem of finding the intersection points of a pencil of lines with rational slope on the \(2\)-dimensional torus. We show that the intersection points belonging to all the lines in the pencil form a finite cyclic group. We also exhibit a generator for this group in terms of the coefficients of the lines. The need for the results presented in this paper arose in dealing with a discrete limited angle model for computerized tomography \((Cf. [3], [5])\).