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 Erdős and Rényi 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.

We consider perfect matchings of the square-octagon lattice, also known as “fortresses” [16]. There is a natural local Markov chain on the set of perfect matchings that is known to be ergodic. However, unlike Markov chains for sampling perfect matchings on the square and hexagonal lattices, corresponding to domino and lozenge tilings, respectively, the seemingly related Markov chain on the squa...

Eddie Cheng, László Lipták, Oakland University and David Sherman∗, Groves High School The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. We find this number for the (n, k)-bubble-sort graphs and classify all the optimal solutions.