Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk − 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank − 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr ≥ 0, stillG − S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.