Extensions to 2-Factors in Bipartite Graphs

A graph is d-bounded if its maximum degree is at most d. We apply the Ore– Ryser Theorem on f -factors in bipartite graphs to obtain conditions for the extension of a 2-bounded subgraph to a 2-factor in a bipartite graph. As consequences, we prove that every matching in the 5-dimensional hypercube extends to a 2-factor, and we obtain conditions for this property in general regular bipartite graphs. For example, to show that every matching in a regular n-vertex bipartite graph extends to a 2-factor, it suffices to show that all matchings with fewer than n/3 edges extend to 2-factors.