Perfect matchings in random graphs with prescribed minimal degree

Abstract

We consider the existence of perfect matchings in random graphs with n vertices (or n + n vertices in the bipartite case) and m random edges, subject to a lower bound on minimum vertex degree. A random bipartite graph without isolated vertices and m n edges with high probability (whp) has a perfect matching iff the average vertex degree is \( 0.5\log n + \log \log n + c_n ,c_n \to \infty \) however slow. A random graph with minimum degree at least two whp has a matching that matches all the vertices except “odd-man-out” vertices, one per each isolated cycle of odd length, and one for the remaining vertex set if its cardinality is odd. So, for n even, whp the random graph has a perfect matching if it does not have isolated odd cycles.