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Keywords:

  • degree sequences;
  • toughness;
  • best monotone theorem

Abstract

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