Abstract
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 ⩾ ξ′(G) (the 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.
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