P-Factor-criticality of Vertex-Transitive Graphs with Large Girth.

A graph of order n is p-factor-critical, where p is an integer with the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. It is well known that a connected vertex transitive graph is 1-factor-critical if it has odd order and is 2-factor-critical or elementary bipartite if it has even order. In this paper, we show that a connected non bipartite vertex-transitive graph G with degree k >= 6 is p-factor-critical, where p is a positive integer less than k with the same parity as its order, if its girth is not less than the bigger one between 6 and k(p-1)+8/2(k-2).