Toughness and the existence of k-factors. IV

In a paper with the same title [3], we proved Chvatal's conjecture thatk-tough graphs havek-factors if they satisfy trivial necessary conditions. In this paper, we prove the following stronger result: Suppose|V(G)| ≥ k + 1,k ⋅ |V(G)| even, and|S| ≥ k ⋅ w(G − S) − 7/8k ifw(G − S) ≥ 2, wherew(G − S) is the number of connected components ofG − S. ThenG has ak-factor.