A new asymptotic enumeration technique: the Lovász Local Lemma

Our previous paper [14] applied a general version of the Lovász Local Lemma that allows negative dependency graphs [11] to the space of random injections from an m-element set to an n-element set. Equivalently, the same story can be told about the space of random matchings in Kn,m. Now we show how the cited version of the Lovász Local Lemma applies to the space of random matchings in K2n. We also prove tight upper bounds that asymptotically match the lower bound given by the Lovász Local Lemma. As a consequence, we give new proofs to results on the enumeration of d-regular graphs. The tight upper bounds can be modified to the space of matchings in Kn,m, where they yield as application asymptotic formulas for permutation and Latin rectangle enumeration problems. As another application, we provide a new proof to the classical probabilistic result of Erdős [8] that showed the existence of graphs with arbitrary large girth and chromatic number. In addition to letting the girth and chromatic number slowly grow to infinity in terms of the number of vertices, we provide such a graph with a prescribed degree sequence, if the degree sequence satisfies some mild conditions. 1 Lovász Local Lemma with negative dependency graphs This is a sequel to our previous paper [14] and we use the same notations. Let A1, A2, . . . , An be events in a probability space. A negative dependency graph forA1, . . . , An is a simple graph on [n] satisfying Pr(Ai| ∧j∈S Aj) ≤ Pr(Ai), (1) This researcher was supported in part by the NSF DMS contract 07