A pair of forbidden subgraphs and perfect matchings in graphs of high connectivity

Sumner [7] proved that every connected K1,3-free graph of even order has a perfect matching. He also considered graphs of higher connectivity and proved that if m ≥ 2, every m-connected K1,m+1-free graph of even order has a perfect matching. In [6], two of the present authors obtained a converse of sorts to Sumner’s result by asking what single graph one can forbid to force the existence of a perfect matching in an m-connected graph of even order and proved that a star is the only possibility. In [2], Fujita et al. extended this work by considering pairs of forbidden subgraphs which force the existence of a perfect matching in a connected graph of even order. But they did not settle the same problem for graphs of higher connectivity. In this paper, we give an answer to this problem. Together with the result in [2], a complete characterization of the pairs is given.