For two vertices \(u\) and \(v\) of a connected graph \(G\), the set \(H(u, v)\) consists of all those vertices lying on a \(u-v\) geodesic in \(G\). Given a set \(S\) of vertices of \(G\), the union of all sets \(H(u,v)\) for \(u,v \in S\) is denoted by \(H(S)\). A convex set \(S\) satisfies \(H(S) = S\). The convex hull \([S]\) is the smallest convex set containing \(S\). The hull number \(h(G)\) is the minimum cardinality among the subsets \(S\) of \(V(G)\) with \([S] = V(G)\). When \(H(S) = V(G)\), we call \(S\) a geodetic set. The minimum cardinality of a geodetic set is the geodetic number \(g(G)\). It is shown that every two integers \(a\) and \(b\) with \(2 \leq a \leq b\) are realizable as the hull and geodetic numbers, respectively, of some graph. For every nontrivial connected graph \(G\), we find that \(h(G) = h(G \times K_2)\). A graph \(F\) is a minimum hull subgraph if there exists a graph \(G\) containing \(F\) as induced subgraph such that \(V(F)\) is a minimum hull set for \(G\). Minimum hull subgraphs are characterized.

For a graph \(G = (V, E)\), a set \(S \subseteq V\) is a dominating set if every vertex in \(V – S\) is adjacent to at least one vertex in \(S\). A dominating set \(S \subseteq V\) is a paired-dominating set if the induced subgraph \(\langle S\rangle\) has a perfect matching. We introduce a variant of paired-domination where an additional restriction is placed on the induced subgraph \(\langle S\rangle \). A paired-dominating set \(S\) is an induced-paired dominating set if the edges of the matching are the induced edges of \(\langle S\rangle\), that is, \(\langle S\rangle\) is a set of independent edges. The minimum cardinality of an induced-paired dominating set of \(G\) is the induced-paired domination number \(\gamma_{ip}(G)\). Every graph without isolates has a paired-dominating set, but not all these graphs have an induced-paired dominating set. We show that the decision problem associated with induced-paired domination is NP-complete even when restricted to bipartite graphs and give bounds on \(\gamma_{ip}(G)\). A characterization of those triples \((a, b, c)\) of positive integers \(a \leq b \leq c\) for which a graph has domination number \(a\), paired-domination number \(b\), and induced-paired domination \(c\) is given. In addition, we characterize the cycles and trees that have induced-paired dominating sets.

Let \(M\) be an \(m\)-subset of \(\mathrm{PG}(k, 2)\), the finite projective geometry of dimension \(k\) over \(\mathrm{GF}(2)\). We would like to know the maximum number of lines that can be contained in \(M\). In this paper, we will not only give the maximum number of lines contained in \(m\)-subsets of \(\mathrm{PG}(k,2)\), but also construct an \(m\)-subset of \(\mathrm{PG}(k,2)\) containing the maximum number of lines.

Maximal partial spreads of the sizes \(13, 14, 15, \ldots, 22\) and \(26\) are described. They were found by using a computer. The computer also made a complete search for maximal partial spreads of size less than or equal to \(12\). No such maximal partial spreads were found.

Suppose we are given a set of sticks of various integer lengths, and that we have a knife that can cut as many as \(w\) sticks at a time. We wish to cut all the sticks up into pieces of unit length. By what procedure should the sticks be cut so that the total number of steps required is minimum? In this paper we show that the following natural algorithm is optimal: at each stage, choose the \(w\) longest sticks (or all sticks of length \(> 1\) if there are fewer than \(w\) of them) and cut them all in half (or as nearly in half as possible).

In this paper, we study intersection assignments of graphs using multiple intervals for each vertex, where each interval is of identical length or in which no interval is properly contained in another. The resulting parameters unit interval number, \(i_u(G)\) and proper interval number, \(i_p(G)\) are shown to be equal for any graph \(G\). Also, \(i_u(G)\) of a triangle-free graph \(G\) with maximum degree \(D\) is \(\left\lceil\frac{D+1}{2}\right\rceil\) if \(G\) is regular and \(\left\lceil\frac{D}{2}\right\rceil\) otherwise.

In [3] Brualdi and Hollingsworth conjectured that for any one-factorization \(\mathcal{F}\) of \(K_n\), there exists a decomposition of \(K_{2n}\) into spanning trees orthogonal to \(\mathcal{F}\). They also showed that two such spanning trees always existed. We construct three such trees and exhibit an infinite class of complete graphs with an orthogonal decomposition into spanning trees with respect to the one-factorization \(GK_{2n}\).

Four generalized theorems involving partitions and \((n+1)\)-color partitions are proved combinatorially. Each of these theorems gives us infinitely many partition identities. We obtain new generating functions for \(F\)-partitions and discuss some particular cases which provide elegant Rogers-Ramanujan type identities for \(F\)-partitions.

The aim of this paper is to study the isoperimetric numbers of double coverings of a complete graph. It turns out that these numbers are very closely related to the bisection widths of the double coverings and the degrees of unbalance of the signed graphs which derive the double coverings. For example, the bisection width of a double covering of a complete graph \(K_m\) is equal to \(m\) times its isoperimetric number. We determine which numbers can be the isoperimetric numbers of double coverings of a complete graph.

A digraph operation called pushing a set of vertices is studied with respect to tournaments. When a set \(X\) of vertices is pushed, the orientation of every arc with exactly one end in \(X\) is reversed. We discuss the problems of which tournaments can be made transitive and which can be made isomorphic to their converse using this operation.