Path and cycle decompositions of complete equipartite graphs: Four parts

Abstract

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 $k\ge 3$, 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 $k+1$.