Minimum degree of minimal (n-10)-factor-critical graphs

Abstract

A graph G of order n is said to be k-factor-critical for integers 1≤k<n, if the removal of any k vertices results in a graph with a perfect matching. A k-factor-critical graph G is called minimal if for any edge e∈E(G), G−e is not k-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimal k-factor-critical graph of order n has minimum degree k+1 and confirmed it for k=1,n−2,n−4 and n−6. By using a novel approach, we have confirmed it for k=n−8 in a previous paper. Continuing with this method, we confirm the conjecture when k=n−10 in this paper.