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 vertical bar V(G)vertical bar 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 for any epsilon > 0, an (n, d, lambda)-graph G (that is, a d-regular graph on n vertices whose second largest eigenvalue in absolute value is at most lambda) admits a 1-factorization provided that n is even, C-0 <= d <= n-1 (where C-0 = C-0(epsilon) is a constant depending only on c), and lambda <= d(1-epsilon). 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.