Path and cycle decompositions of complete equipartite graphs: Four parts

We show that a complete equipartite graph with four partite sets has an edge-disjoint decomposition into cycles of length k if and only if k ≥ 3 , the partite set size is even, k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k . We also show that a complete equipartite graph with four even partite sets has an edge-disjoint decomposition into paths with k edges if and only if k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k + 1 .