A Normal Law for Matchings

Dedicated to the memory of Paul Erdős, both for his pioneering discovery of normality in unexpected places, and for his questions, some of which led (eventually) to the present work.

For a simple graph G, let be the size of a matching drawn uniformly at random from the set of all matchings of G. Motivated by work of C. Godsil [11], we give, for a sequence and , several necessary and sufficient conditions for asymptotic normality of the distribution of , for instance

(where E and is the size of a largest matching in ). In particular this gives asymptotic normality for any sequence of regular graphs (of positive degree) or graphs with perfect matchings. When tends to a finite limit, a sufficient (and probably necessary) condition is given for to be asymptotically Poisson.

The material presented here suggests numerous related questions, some of which are discussed in the last section of the paper.