Minimally k-Factor-Critical Graphs for Some Large k

Abstract

A graph G of order n is said to be k-factor-critical for integers $$1\le k < n$$ , if the removal of any k vertices results in a graph with a perfect matching. 1- and 2-factor-critical graphs are the well-known factor-critical and bicritical graphs, respectively. A k-factor-critical graph G is called minimal if for any edge $$e\in E(G)$$ , $$G-e$$ is not k-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimally k-factor-critical graph of order n has the minimum degree $$k+1$$ and confirmed it for $$k=1, n-2, n-4$$ and $$n-6$$ . In this paper, we use a simple method to reprove the above results. As a main result, the further use of this method enables us to prove the conjecture to be true for $$k=n-8$$ . We also obtain that every minimally $$(n-6)$$ -factor-critical graph of order n has at most $$n-\Delta (G)$$ vertices with the maximum degree $$\Delta (G)$$ for $$\Delta (G)\ge n-4$$ .