## Abstract

Let G = G(n) be a randomly chosen k-edge-coloured k-regular graph with 2n vertices, where k = k(n). Such a graph can be obtained from a random set of k edge-disjoint perfect matchings of K_{2n}. Let h = h(n) be a graph with m = m(n) edges such that m^{2} + 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.

Original language | English |
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Pages (from-to) | 533-549 |

Number of pages | 17 |

Journal | Combinatorics Probability and Computing |

Volume | 18 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2009 |