An Ore-type condition for hamiltonicity in tough graphs and the extremal examples

Abstract

Let G be a t-tough graph on n ≥ 3 vertices for some t > 0. It was shown by Bauer et al. in 1995 that if the minimum degree of G is greater than n t+1 − 1, then G is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when t is between 1 and 2, and recently the author proved a general result. The result states that if the degree sum of any two nonadjacent vertices of G is greater than 2n t+1 + t − 2, then G is hamiltonian. It was conjectured in the same paper that the "+t" in the bound 2n t+1 + t − 2 can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer et al.. Furthermore, we characterize all t-tough graphs G on n ≥ 3 vertices for which σ 2 (G) = 2n t+1 − 2 but G is non-hamiltonian.