4-Factor-criticality of vertex-transitive graphs

TL;DR

This paper characterizes when connected non-bipartite vertex-transitive graphs of even order are 4-factor-critical, linking this property to the graph's degree, and applies findings to Cayley graphs to show 2-extendability.

Contribution

It establishes a precise degree condition for 4-factor-criticality in non-bipartite vertex-transitive graphs of even order, advancing understanding of their matching properties.

Findings

Connected non-bipartite vertex-transitive graphs of even order ≥6 are 4-factor-critical iff degree ≥5.

Such graphs with degree ≥5 are 2-extendable if they are Cayley graphs.

Provides a characterization linking degree and factor-criticality in these graphs.

Abstract

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.