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 ∈ 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. [ABSTRACT FROM AUTHOR]