# 1-factorizations of pseudorandom graphs

A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). A trivial necessary condition for G to admit a 1-factorization is that |V(G)| is even and G is regular; the converse is easily seen to be false. In this paper, we consider the problem of finding 1-factorizations of regular, pseudorandom graphs. Specifically, we prove that an (n,d,λ)-graph G (that is, a d-regular graph on n vertices whose second largest eigenvalue in absolute value is at most λ) admits a 1-factorization provided that n is even, C_0≤ d≤ n-1 (where C_0 is a universal constant), and λ≤ d^1-o(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 off by a factor of 2 in the base of the exponent from the known upper bound. This lower bound 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. Our proofs are probabilistic and can be easily turned into polynomial time (randomized) algorithms.

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