Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs

TL;DR

This paper investigates the emergence of perfect matchings in random subgraphs of regular bipartite graphs, revealing a threshold phenomenon depending on the degree of regularity.

Contribution

It extends classical results to all sufficiently large degrees in regular bipartite graphs and shows exceptions for smaller degrees.

Findings

Perfect matchings typically appear when the last isolated vertex disappears for large degrees.

Existence of bipartite regular graphs where perfect matchings appear much later than the last isolated vertex.

Threshold behavior depends on the degree k relative to n and log factors.

Abstract

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary $k$-regular bipartite graphs $G$ on $2n$ vertices for all $k = \omega \left( \frac{n}{\log^{1/3} n} \right)$. Surprisingly, this is not the case for smaller values of $k$. Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite $k$-regular graphs in which the last isolated vertex disappears long before a perfect matching appears.