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Subgraphs of Random k-Edge-Coloured k-Regular Graphs

Published online by Cambridge University Press:  01 July 2009

PAULETTE LIEBY ,
BRENDAN D. McKAY ,
JEANETTE C. McLEOD  and
IAN M. WANLESS
PAULETTE LIEBY
Affiliation:
National ICT Australia, Canberra Research Laboratory, Locked Bag 8001, Canberra ACT 2601, Australia (e-mail: [email protected])
BRENDAN D. McKAY
Affiliation:
School of Computer Science, Australian National University, Canberra ACT 0200, Australia (e-mail: [email protected])
JEANETTE C. McLEOD
Affiliation:
Heilbronn Institute, Department of Mathematics, University of Bristol, Bristol BS8 1TH, UK (e-mail: [email protected])
IAN M. WANLESS
Affiliation:
School of Mathematical Sciences, Monash University, Vic 3800, Australia (e-mail: [email protected])
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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 K2n. 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(n5/6) is found.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 4 , July 2009 , pp. 533 - 549
Copyright
Copyright © Cambridge University Press 2009

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