Toughness and Vertex Degrees

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t ‐tough. We first give a best monotone theorem when t ≥ 1 , but then show that for any integer k ≥ 1 , a best monotone theorem for t = 1 k ≤ 1 requires at least f ( k ) · | V ( G ) | nonredundant conditions, where f ( k ) grows superpolynomially as k → ∞ . When t < 1 , we give an additional, simple theorem for G to be t ‐tough, in terms of its vertex degrees.