Elizabeth J. Billington

Decompositions of Complete Multipartite Graphs into Cycles of Even Length

Abstract. Necessary and sufficient conditions for the existence of an edge-disjoint decomposition of any complete multipartite graph into even length cycles are investigated. Necessary conditions are listed and sufficiency is shown for the cases when the cycle length is 4, 6 or 8. Further results concerning sufficiency, provided certain “small” decompositions exist, are also given for arbitrary even cycle lengths.

On the Hamilton-Waterloo problem

Abstract. The Hamilton-Waterloo problem asks for a 2-factorisation of Kv in which r of the 2-factors consist of cycles of lengths a1,a2,…,at and the remaining s 2-factors consist of cycles of lengths b1,b2,…,bu (where necessarily ∑i=1tai=∑j=1ubj=v). In this paper we consider the Hamilton-Waterloo problem in the case ai=m, 1≤i≤t and bj=n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}.

Combinatorial trades: a survey of recent results

The concept of a trade in a combinatorial structure has existed for some years now. However, in the last five years or so there has been a great deal of activity in the area. This survey paper builds upon the one by Khosrovshahi, Maimani and Torabi which appeared in Discrete Applied Mathematics (Volume 95, pp. 361-176) in 1999. In the short time since that survey appeared, the number of papers in the area has almost doubled. Trades are used in designs and latin squares; they also crop up in graph theory. In this paper the most recent work on trades is surveyed, with applications given.

The three-way intersection problem for latin squares

The set of integers k for which there exist three latin squares of order n having precisely k cells identical, with their remaining n(2) - k cells different in all three latin squares, denoted by I-3[n], is determined here for all orders n. In particular, it is shown that I-3[n] = {0,...,n(2) - 15} {n(2) - 12,n(2) - 9,n(2)} for n greater than or equal to 8

Packing complete multipartite graphs with 4-cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.

Decomposition of complete tripartite graphs into gregarious 4-cycles

A 4-cycle in a tripartite graph with vertex partition {V1, V2, V3} is said to be gregarious if it has at least one vertex in each Vi, 1 ≤ i ≤ 3. In this paper, necessary and sufficient conditions are given for the existence of an edge-disjoint decomposition of any complete tripartite graph into gregarious 4-cycles.

Trades and Graphs

Abstract. The trade spectrum of a simple graph G is defined to be the set of all t for which it is possible to assemble together t copies of G into a simple graph H, and then disassemble H into t entirely different copies of G. Trade spectra of graphs have applications to intersection problems, and defining sets, of G-designs. In this investigation, we give several constructions, both for specific families of graphs, and for graphs in general.

Maximum packings of uniform group divisible triple systems

Necessary and sufficient conditions are given for the existence of a maximum packing of K with triangles with all possible minimal leaves. This is of course equivalent to a maximum packing of a group divisible triple system with n groups of size m.

The spectrum for lambda-fold 2-perfect 6-cycle systems

Abstract The spectrum for the decomposition ofλKv into 2-perfect 6-cycle systems is found for allλ > 1 and for two outstanding cases whenλ = 1, completing work done by Lindner, Phelps and Rodger in the caseλ = 1.

Path and cycle decompositions of complete equipartite graphs: Four parts

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k-cycles, Australas. J. Combin. 18 (1998) 193-200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m, has an edge-disjoint decomposition into paths of length k if and only if k divides 3m^2 and k =3 if and only if k divides 10m^2 and k=<5m for cycles (or k<5m for paths).