Matchings in Random Biregular Bipartite Graphs

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.Comment: 30 pages and 3 figures - Latest version has updated introduction and bibliograph

A superlinear bound on the number of perfect matchings in cubic bridgeless graphs

Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde