Isomorphic factorization of r-regular graphs into r parts

AbstractA 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