A pair of forbidden subgraphs and perfect matchings

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to be H -free if no graph in H is an induced subgraph of G. We completely characterize the set H such that every connected H -free graph of sufficiently large even order has a perfect matching in the following cases. graph in H is triangle-free. sists of two graphs (i.e. a pair of forbidden subgraphs). hing M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H -free graph of sufficiently large odd order has a near-perfect matching in the above cases.