Toughness and the existence of k-factors. IV Original Research Article

For an undirected graph G, a variation of toughness is defined asτ(G)≔min|S|w(G−S)−1w(G−S)⩾2if G is not complete, and τ(G)≔∞ if G is complete, where w(G−S) is the number of connected components of G−S. In a paper with the same title, we proved that when k=1 or 2, G has a k-factor if τ(G)⩾k and G satisfies trivial necessary conditions. This is not true for k⩾3. However, there are only finitely many exceptions for each k. More precisely, we prove that G has a k-factor if τ(G)⩾k, k·|G| even, and |G|⩾k2−1. As an application of this result, we can solve a problem posed by Cai et al.: Suppose t(G)⩾k, t(G)>(s+k−1)/2, k·(|G|+s) even, and |G|⩾s+k2−1, where t(G) is the toughness of G. Then G is (k,s)-factor-critical, that is, G−S has a k-factor for any subset S of V(G) with |S|=s.