Abstract

AbstractConsider the random process in which the edges of a graph are added one by one in a random order. A classical result states that if is the complete graph or the complete bipartite graph , then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary ‐regular bipartite graphs on vertices for all . Surprisingly, this is not the case for smaller values of . Using a construction due to Goel, Kapralov, and Khanna, we show that there exist bipartite ‐regular graphs in which the last isolated vertex disappears long before a perfect matching appears.

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