# The ACO Seminar (2018–2019)

October 11, 3:30pm, Wean 8220
Michael Simkin, Hebrew University of Jerusalem
Perfect matchings in random subgraphs of regular bipartite graphs

Abstract:

The classical theory of Erdős–Rényi random graphs is concerned primarily with random subgraphs of $K_n$ or $K_{n,n}$. Lately, there has been much interest in understanding random subgraphs of other graph families, such as regular graphs.

We study the following problem: Let G be a k-regular bipartite graph with 2n vertices. Consider the random process where, beginning with 2n isolated vertices, G is reconstructed by adding its edges one by one in a uniformly random order. An early result in the theory of random graphs states that if $G=K_{n,n}$, then with high probability a perfect matching appears at the same moment that the last isolated vertex disappears. We show that if $k=\Omega(n)$ then this holds for any k-regular bipartite graph G. This improves on a result of Goel, Kapralov, and Khanna, who showed that with high probability a perfect matching appears after $O(n \log(n))$ edges have been added to the graph. On the other hand, if $k = o(n / (\log(n) \log \log(n)))$, we construct a family of k-regular bipartite graphs in which isolated vertices disappear long before the appearance of perfect matchings.

Joint work with Roman Glebov and Zur Luria

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.