A new asymptotic enumeration technique: the Lovasz Local Lemma

Abstract

Our previous paper applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs 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 $K_{n,m}$. Now we show how the cited version of the Lovász Local Lemma applies to the space of random matchings in $K_{2n}$. 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 $K_{n,m}$, where they yield as application asymptotic formulas for permutation and Latin rectangle enumeration problems. The strength of the method is shown by a new result: enumeration of graphs by degree sequence or bipartite degree sequence and girth. As another application, we provide a new proof to the classical probabilistic result of Erd\H os that showed the existence of graphs with arbitrary large girth and chromatic number. If the degree sequence satisfies some mild conditions, almost all graphs with this degree sequence and prescribed girth have high chromatic number.