Generalized matching extension problem in regular bipartite graphs

Let m and r be integers with r ≥ 3 , let G be a cyclically ( m r − r + 1 ) -edge-connected r -regular bipartite graph and let M be a matching in G of size m . Plummer (1991) proved that, if r ≥ m + 1 , then there exists a perfect matching of G containing M . More recently, Aldred et al. (2023) proved that if m ≥ r and no vertex of G is adjacent to r − 1 vertices of V ( M ) , then G has a perfect matching containing M . In these theorems, if we let H ≃ m K 2 be the subgraph of G induced by the edge set of M , then the conclusion is equivalent to the statement that G − H has a perfect matching. In this paper, we generalize these results to the case where H is a subgraph of G other than m K 2 , and we prove that the same conclusion holds whenever H is a balanced bipartite subgraph of G with 2 m vertices. Moreover, we show that if | E ( H ) | > 1 2 m r , then the condition on cyclic edge-connectivity can be weakened to 2 m r − 2 | E ( H ) | − r + 1 .