Benjamin R. Smith

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).

Decompositions of complete multigraphs into cycles of varying lengths

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

Equipartite Gregarious 5-Cycle Systems and Other Results

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph Km(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).

Decomposing Complete Equipartite Graphs into Closed Trails of Length k

Necessary conditions for a simple connected graph G to admit a decomposition into closed trails of length k ≥ 3 are that G is even and its total number of edges is a multiple of k. In this paper we show that these conditions are sufficient in the case when G is the complete equipartite graph having at least three parts, each of the same size.

Face 2-colorable embeddings with faces of specified lengths

Suppose M=m1,m2,.,mr and N=n1,n2,.,nt are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2-colorable planar embedding in which the faces of one color have boundary lengths m1,m2,.,mr and the faces of the other color have boundary lengths n1,n2,.,nt. Such a graph is said to have a planar (M;N)-biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths m1,m2,.,mr and also into t cycles of lengths n1,n2,.,nt. Such a graph is said to be (M;N)-decomposable.