Nicholas J. Cavenagh

The theory and application of latin bitrades: A survey

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same sets of symbols. This survey paper summarizes the theory of latin bitrades, detailing their applications to critical sets, random latin squares and existence constructions for latin squares.

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.

3-Homogeneous latin trades

Let T be a partial latin square and I be a latin square with T ⊆ L. We say that T is a latin trade if there exists a partial latin square T′ with T′ ∩ T = 0 such that (L\T) ∪ T′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 4-homogeneous latin trades from rectangular packings of the plane with circles.

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).

Further decompositions of complete tripartite graphs into 5-cycles

Let K(r,s,t) denote the complete tripartite graph with partite sets of sizes r, s and t, where r ≤ s ≤ t. Necessary and sufficient conditions are given for decomposability of K(r,s,t) into 5-cycles whenever r,s and t are all even. This extends work done by Mahmoodian and Mirzakhani (Decomposition of complete tripartite graphs into 5-cycles, in: Combinatorics Advances, Kluwer Academic Publishers, Netherlands, 1995, pp. 235-241) and Cavenagh and Billington (J. Combin. 22 (2000) 41).

Minimal homogeneous latin trades

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d>=3, and m>=1.75d^2+3. We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct.

Nonextendible Latin Cuboids

We show that for all integers

Latin bitrades derived from groups

m \geqslant 4

On a generalization of the Oberwolfach problem

there exists a

Constructing and deconstructing latin trades

2m\times 2m\times m