Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into kk-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 kk. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size mm, has an edge-disjoint decomposition into paths of length kk if and only if kk divides 3m23m2 and k<3mk<3m. Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size mm has an edge-disjoint decomposition into cycles (and also into paths) of length kk with k⩾3k⩾3 if and only if kk divides 10m210m2 and k⩽5mk⩽5m for cycles (or k<5mk<5m for paths).