Common multiples of paths and stars with complete graphs

Abstract

A graph G is a common multiple of two graphs H1 and H2 if there exists a decomposition of G into edge-disjoint copies of H1 and also a decomposition of G into edge-disjoint copies of H2. If G is a common multiple of H1 and H2 and G has q edges, then we call G a (q, H1, H2) graph. Our paper deals with the following question: ''Given two graphs H1 and H2, for which values of q does there exist a (q, H1, H2) graph?'' when H1 is either a path or a star with 3 or 4 edges and H2 is a complete graph.