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

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 is an element of 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.(c) 2023 Elsevier B.V. All rights reserved.