A result on decompositions of regular graphs

Su, X.-Y., A result on decompositions of regular graphs, Discrete Mathematics 105 (1992) 323-326. We prove that for any connected graph G and any integer r which is a common multiple of the degrees of the vertices in G, there exists a connected, r-regular, and G-decomposable graph H such that x(H) = x(G) and o(H) = w(G), where x and w are the chromatic number and the clique number, respectively. Also we give a bound for the minimum order among all such graphs. Only simple graphs are considered. Given a graph G, a graph H is called G-decomposable if there exists a partition of E(H) into disjoint subsets E(G,) such that each of the graphs G, induced by E(G,) is isomorphic to the graph G. Wilson [2] has shown that for any graph G, the complete graph K, is G-decomposable for n sufficiently large if the following obvious conditions hold: n(n 1)/2 is divisible by IE(G)I an d n 1 is a multiple of the greatest common divisor of the degrees of the vertices of G. Recently, Fink [l] introduced a new parameter r,,(G) for any connected regular graph G. Theorem 1. For any connected graph G and any integer r which is a common multiple of the degrees of the vertices in G, there exists a connected, r-regular, and G-decomposable graph H such that x(H) = x(G) and o(H) = o(G), where x and o are the chromatic number and the clique number, respectively. Proof. Let G be any connected graph and let r be any integer which is a common 0012-365X/92/$05.00 @ 1992 Elsevier Science Publishers B.V. All rights reserved