1‐Factorizations of pseudorandom graphs

Abstract

A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). In this paper, we prove that for any > 0, an (n, d, λ)-graph G admits a 1-factorization provided that n is even, C 0 ≤ d ≤ n − 1 (where C 0 = C 0 ( ) is a constant depending only on ), and λ ≤ d 1− . In particular, since (as is well known) a typical random d-regular graph G n,d is such a graph, we obtain the existence of a 1-factorization in a typical G n,d for all C 0 ≤ d ≤ n−1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G, which is better by a factor of 2 nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.