The independence number \(\beta_n\), for knights on equilateral triangular boards \(T_n\), of regular hexagons is determined for all \(n\).

It was conjectured by Lee that a cubic simple graph with \(4k + 2\) vertices is edge-magic [5]. In this paper we show that the conjecture is not true for multigraphs or disconnected simple graphs in general. Several new classes of cubic edge-magic graphs are exhibited.

In 1976 Erdős asked about the existence of Steiner triple systems that lack collections of \(j\) blocks employing just \(j+2\) points. This has led to the study of anti-Pasch, anti-mitre and 5-sparse Steiner triple systems. Simultaneously generating sets and bases for Steiner triple systems and \(t\)-designs have been determined. Combining these ideas, together with the observation that a regular graph is a 1-design, we arrive at a natural definition for the girth of a design. In turn, this provides a natural extension of the search for cages to the universe of all \(t\)-designs. We include the results of computational experiments that give an abundance of examples of these new definitions.

A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(x, y,\) and \(z\) with \(d(x,y) = 2\) and \(z \in N(x) \cap N(y)\), \(d(x) + d(y) \geq |N(x) \cup N(y) \cup N(z)| – 1\). Let \(G\) be a \(3\)-connected \(L_1\)-graph of order \(n \geq 18\). If \(\delta(G) \geq n/3\), then every pair of vertices \(u\) and \(v\) in \(G\) with \(d(u,v) \geq 3\) is connected by a Hamiltonian path of \(G\).

How many vertices must we delete from a graph so that it no longer contains a path \(P_k\) on \(k\) vertices? We explore this question for various special graphs (hypercubes, square lattice graphs) as well as for some general families.

A complete list is given of all finite trivalent arc-transitive connected graphs on up to \(768\) vertices, completing and extending the Foster census. Several previously undiscovered graphs appear, including one on \(448\) vertices which is the smallest arc-transitive trivalent graph having no automorphism of order 2 which reverses an arc. The graphs on the list are classified according to type (as described by Djokovic and Miller in terms of group amalgams), and were produced with the help of a parallel program which finds all normal subgroups of low index in a finitely-presented group. Further properties of each graph are also given: its girth, diameter, Hamiltonicity, and whether or not it is bipartite.

In this paper the decomposition of Dyck words into a product of Dyck prime subwords is studied. The set of Dyck words which are decomposed into \(k\) components is constructed and its cardinal number is evaluated.

For an ordered set \(W = \{w_1, w_2, \ldots, w_k\}\) of vertices and a vertex \(v\) in a graph \(G\), the representation of \(v\) with respect to \(W\) is the \(k\)-vector \(r(v|W) = (d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\), where \(d(x,y)\) represents the distance between the vertices \(x\) and \(y\). The set \(W\) is a resolving set for \(G\) if distinct vertices of \(G\) have distinct representations. A resolving set containing a minimum number of vertices is called a basis for \(G\) and the number of vertices in a basis is the (metric) dimension \(\dim G\). A connected graph is unicyclic if it contains exactly one cycle. For a unicyclic graph \(G\), tight bounds for \(\dim G\) are derived. It is shown that all numbers between these bounds are attainable as the dimension of some unicyclic graph.

It is an established fact that some graph-theoretic extremal questions play an important part in the investigation of communication network vulnerability. Questions concerning the realizability of graph invariants are generalizations of the extremal problems. We define a \((p,q, \kappa,\delta)\) graph as a graph having \(p\) vertices, \(q\) edges, vertex connectivity \(\kappa\) and minimum degree \(\delta\). An arbitrary quadruple of integers \((a,b, c, d)\) is called \((p,q, \kappa, \delta)\) realizable if there is a \((p,q, \kappa, \delta)\) graph with \(p=a, q=b, \kappa=c\) and \(\delta=d\). Necessary and sufficient conditions for a quadruple to be \((p,q, \kappa, \delta)\) realizable are derived. In earlier papers, Boesch and Suffel gave necessary and sufficient conditions for \((p,q, \kappa), (p,q, \lambda), (p,4, \delta), (p, \Delta,\delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability, where \(\Delta\) denotes the maximum degree for all vertices in a graph and \(\lambda\) denotes the edge connectivity of a graph.

Let \(\lambda DK_v\). denote the complete directed multigraph with \(v\). vertices, where any two distinct vertices \(x\). and \(y\). are joined by \(\lambda\). arcs \((x,y)\). and \(\lambda\). arcs \((y,x)\).. By a \(k\).-circuit we mean a directed cycle of length \(k\).. In this paper, we consider the problem of constructing maximal packings and minimal coverings of \(\lambda DK_v\). with \(k\).-circuits. Using the leave-arcs graph of packing and the repeat-arcs graph of covering, we give a unified method for finding packings and coverings. Also, we completely solve the existence of optimal packings and coverings for \(5 \leq k \leq 14\). and any \(\lambda\).