Subgraphs of random k-edge-coloured k-regular graphs

Let G=G(n) be a randomly chosen k-edge-coloured k-regular graph with 2n vertices, where k=k(n). Equivalently, G is the union of a random set of k disjoint perfect matchings. Let h=h(n) be a graph with m=m(n) edges such that m²+mk=o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs of G isomorphic to h. Isomorphisms may or may not respect the edge colouring, and other generalizations are also presented. Special attention is paid to matchings and cycles. The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number of k-edge-coloured k-regular graphs for k=o(n^5/6) is found.

Last modified: Tue Jan 23 15:29:26 EST 2007