Subgraphs of Randomk-Edge-Colouredk-Regular Graphs

Abstract

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 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 generalisations 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.

Subgraphs of Random Graphs with Specified Degrees

“…As first shown by McKay and Wormald [26] in a slightly different context, the counting is substantially easier if the more complex switching operation of Figure 3 is used. The other improvement, introduced by Lieby, McKay, McLeod and Wanless [18], is a rearrangement of the calculation. Let the edges of X be e 1 , e 2 , .…”

Functional limit theorems for random regular graphs

“…See also [LMMW09,Theorem 4.1], in which the authors consider a different model of random regular graph and find that the limiting mean number of cycles of length k differs slightly from both of these. Next we consider the number of short non-backtracking walks on the graph; a non-backtracking walk is a closed walk that never follows an edge and immediately retraces that same edge backwards.…”

“…See also [LMMW09,Theorem 4.1], in which the authors consider a different model of random regular graph and find that the limiting mean number of cycles of length k differs slightly from both of these.…”

Subgraph statistics in subcritical graph classes

“…Since then, lot of effort has been devoted to locate the threshold function for the appearance of a given subgraph in the G(n, p) model, as well as the limiting distribution of the corresponding counting random variable (see for instance [27,25,39], and the monograph [26,Chapter 3]). The number of appearances of a fixed graph and its statistics had been also addressed as well in different restricted graph classes, including random regular graphs and random graphs with specified vertex degree (see for instance, [31,17,29,28,33], see also [32]) and random planar maps [18,19].…”