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Our previous paper applied a lopsided version of the Lov\'asz Local Lemmathat allows negative dependency graphs to the space of random injections froman $m$-element set to an $n$-element set. Equivalently, the same story can betold about the space of random matchings in $K_{n,m}$. Now we show how thecited version of the Lov\'asz Local Lemma applies to the space of randommatchings in $K_{2n}$. We also prove tight upper bounds that asymptoticallymatch the lower bound given by the Lov\'asz Local Lemma. As a consequence, wegive new proofs to results on the enumeration of $d$-regular graphs. The tightupper bounds can be modified to the space of matchings in $K_{n,m}$, where theyyield as application asymptotic formulas for permutation and Latin rectangleenumeration problems. The strength of the method is shown by a new result:enumeration of graphs by degree sequence or bipartite degree sequence andgirth. As another application, we provide a new proof to the classicalprobabilistic result of Erd\H os that showed the existence of graphs witharbitrary large girth and chromatic number. If the degree sequence satisfiessome mild conditions, almost all graphs with this degree sequence andprescribed girth have high chromatic number.

A new asymptotic enumeration technique: the Lovasz Local Lemma