  A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the
same parity as $n$, if the removal of any set of $p$ vertices results in a
graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical
graphs are well-known factor-critical graphs and bicritical graphs,
respectively. It is known that if a connected vertex-transitive graph has odd
order, then it is factor-critical, otherwise it is elementary bipartite or
bicritical. In this paper, we show that a connected vertex-transitive
non-bipartite graph of even order at least 6 is 4-factor-critical if and only
if its degree is at least 5. This result implies that each connected
non-bipartite Cayley graphs of even order and degree at least 5 is
2-extendable.
