  Our previous paper applied a lopsided version of the Lov\'asz 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 Lov\'asz 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 Lov\'asz 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.
