Isomorphic factorization of regular graphs and 3-regular multigraphs

A multigraph G is divisible by t if its edge set can be partitioned into t 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 t does not divide e(G). A short proof is given that any graph G is t-rational for all t ≥ x’(G) (t n e chromatic index of G), and thus any r-regular graph is t-rational for all t ≥ 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 t-rational for all t ≥ 3. The proofs rely on edge-colouring techniques.