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