Isomorphic factorization of r-regular graphs into r parts

Abstract A graph 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. It is proved that an r-regular graph G with an even number of vertices v(G) is divisible by r, in such a way that the components of each factor are paths of length 1 and 2, under any of the following conditions: 1. (a) r⩾9 is odd and v(G)> 32(r+1) (r−7) ; 2. (b) r⩾18 and v(G)> 24(r+1) (r−17) ; 3. (c) r=25 or 27, or r⩾29.