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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 math formula, but then show that for any integer math formula, a best monotone theorem for math formula requires at least math formula nonredundant conditions, where math formula grows superpolynomially as math formula. When math formula, we give an additional, simple theorem for G to be t-tough, in terms of its vertex degrees.