Path and cycle decompositions of complete equipartite graphs: 3 and 5 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 3 m 2 and k < 3 m . Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size m has an edge-disjoint decomposition into cycles (and also into paths) of length k with k ⩾ 3 if and only if k divides 10 m 2 and k ⩽ 5 m for cycles (or k < 5 m for paths).