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

TL;DR

This paper proves a conjecture that relaxes Ore-type conditions for hamiltonicity in tough graphs, providing a broader criterion and characterizing extremal non-hamiltonian cases.

Contribution

It confirms the conjecture that the '+t' term can be removed from the Ore-type hamiltonicity condition in tough graphs and characterizes extremal non-hamiltonian graphs.

Findings

The '+t' term in the degree sum condition can be omitted.

A complete characterization of certain extremal tough graphs is provided.

The results generalize previous hamiltonicity conditions for tough graphs.

Abstract

Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, 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 $\frac{2n}{t+1}+t-2$, then $G$ is hamiltonian. It was conjectured in the same paper that the ``$+t$" in the bound $\frac{2n}{t+1}+t-2$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $t$-tough graphs $G$ on $n\ge 3$ vertices for which $\sigma_2(G) = \frac{2n}{t+1}-2$ but $G$ is non-hamiltonian.