Isomorphic Factorization of Regular Graphs and 3-regular Multigraphs

A multigraph G is divisible by t if its edge set can be partitioned into / subsets, such that the subgraphs (called factors) induced by the subsets are all isomorphic. If G has e(G) edges, then it is t-rational if it is divisible by t or if / does not divide e{G). A short proof is given that any graph G is /-rational for all t ^ x'(G) ( t n e chromatic index of G), and thus any r-regular graph is /-rational for all / ^ r + 1. The main result of this paper is that all 3-regular multigraphs are divisible by 3, in such a way that the components of each factor are paths of length 1 or 2. It follows that 3-regular graphs are /-rational for all / ^ 3. The proofs rely on edge-colouring techniques.