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.

Let \(I(G)\) be a graphical invariant defined for any graph \(G\). For several choices of \(I\) representing domination parameters, we characterize sequences of positive integers \(a_1,a_2,\ldots,a_n\) which have an associated sequence of graphs \(G_1,G_2,\ldots,G_n\) such that \(G_i\) has \(i\) vertices, \(G_i\) is an induced subgraph of \(G_{i+1}\), and \(I(G_i) = a_i\).

The fine structure of a directed triple system of index \(\lambda\) is the vector \((c_1,c_2,\ldots,c_\lambda)\), where \(c_i\) is the number of directed triples appearing precisely \(i\) times in the system. We determine necessary and sufficient conditions for a vector to be the fine structure of a directed triple system of index \(3\) for \(v \equiv 2 \pmod{3}\).

We show that if, for any fixed \(r\), the neighbourhood unions of all \(r\)-sets of vertices are large enough, then \(G\) will have many edge-disjoint perfect matchings. In particular, we show that given fixed positive integers \(r\) and \(c\) and a graph \(G\) of even order \(n\), if the minimum degree is at least \(r + c – 1\) and if the neighbourhood union of each \(r\)-set of vertices is at least \(n/2 + \left(2\lfloor\frac{(c + 1)}{2}\rfloor – 1\right)r\), then \(G\) has \(c\) edge-disjoint perfect matchings, for \(n\) large enough. This extends earlier work by Faudree, Gould and Lesniak on neighbourhood unions of pairs of vertices.