W. D. Wallis

Combinatorics : room squares, sum-free sets, Hadamard matrices

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On one-factorizations of complete graphs

We use standard graph notation and definitions, as in [1]: in particular K n is the complete graph on n vertices and K n, n is the regular complete bigraph of order 2 n .

Two-fold triple systems without repeated blocks

Direct, easy, and self-contained proofs are presented of the existence of triple systems with @l = 2 having no repeated blocks.

A Note on One-Factorizations Having a Prescribed Number of Edges in Common

Publisher Summary This chapter presents a note on one-factorizations having a prescribed number of edges in common. A 1-factor of the finite set V is just a set of 2-element subsets of V which partition V . In the chapter, the 2-element subsets of V will be called “edges.” A l-factorization is a pair ( V, F ) where V is a finite set and F is a collection of 1-factors of V , which partition ( v/2) , the set of all edges of V . F is a 1-factorization of V . The number | V | is called the “order of the 1-factorization ( V , F ),” and the spectrum for 1-factorizations is precisely the set of all even positive integers. The chapter presents a complete solution to the intersection problem for 1-factorization. The chapter explains the sets J [ υ ] for small υ.

On one‐factorizations of compositions of graphs

Let G[H] denote the composition of the graphs G and H. If G can be decomposed into one-factors and two-factors, H can be decomposed into one-factors, and H is not the empty graph on an odd number of vertices, then G[H] can be decomposed into one-factors.

SUM-FREE SETS, COLOURED GRAPHS AND DESIGNS

Sum-free sets may be used to colour the edges of a complete graph in such a way as to avoid monochromatic triangles. We discuss the automorphism groups of such graphs. Embedding of colourings is considered. Finally we illustrate a way of constructing colourings using block designs.

The Clique Partition Number of the Complement of a Cycle

We consider the problem of determining the clique partition number of the complement K n - C n of a cycle C n . A complete set of lower bounds is constructed.

SOME NEW ORTHOGONAL DIAGONAL LATIN SQUARES

A recursive construction for orthogonal diagonal latin squares, using group divisible designs, is presented. In consequence the numbers of orders for which the existence of such squares is in question is reduced to 72.

On clique partitions of split graphs

Abstract Split graphs are graphs formed by taking a complete graph and an empty graph disjoint from it and some or all of the possible edges joining the two. We prove that the problem of deciding the clique partition number is NP-complete, even when restricted to the class of split graphs.

Maximal-clique partitions of interval graphs

Abstract It is shown that if an interval graph possesses a maximal-clique partition then its clique coveringand clique partition numbers are equal, and equal to the maximal-clique partition number.Moreover an interval graph has such a partition if and only if all its maximal cliques areedge-disjoint .1980 Mathematics subject classification (Amer. Math. Soc): 05 C 35. 1. Interval graphs and clique-matricesThroughout this paper graphs are finite, undirected, loopless and without mul-tiple edges. A clique is a complete subgraph, and a maximal clique is a cliquewhich is not a proper subgraph of any other clique.A graph G is called an interval graph if its vertices can be put into one-to-onecorrespondence with a set of intervals 7 of the real line, such that two verticesare connected by an edge of G if and only if the corresponding intervals havenonempty intersection. Clearly any induced subgraph of an interval graph is aninterval graph.The earliest characterization of interval graphs was obtained by Lekkerkerkerand Boland [3], as follows.