A recursive theorem on matching extension

A graph G having a perfect matching (or I-factor) is called n-fextendable if every matching of size n is extended to a I-factor. Further, G is said to be 〈r: m, n 〉-extendable if, for every connected subgraph S of order 2r for which G \ V(S) is connected, S is m-extendable and G \ V(S) is nextendable. We prove the following: Let p, r, m, and n be positive integers with p - r > nand r > m. Then every 2-connected 〈r: m, n〉-extendable graph of order 2p is 〈r + 1: m + 1, n - 1〉-extendable.