\(p\)-Factor-Criticality of Vertex-Transitive Graphs with Large Girth

Abstract

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 \(k \geq 6\) 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 \( \frac{k(p-1)+8}{2(k-2)}\).