An Ore-type condition for hamiltonicity in tough graphs

TL;DR

This paper establishes a new Ore-type degree sum condition for Hamiltonicity in t-tough graphs, extending previous degree conditions to a broader range of toughness values.

Contribution

It introduces a novel Ore-type condition for Hamiltonicity in t-tough graphs, generalizing earlier degree sum criteria for specific toughness ranges.

Findings

Proves that degree sum > (2n)/(t+1) + t - 2 guarantees Hamiltonicity in t-tough graphs.

Extends Ore-type Hamiltonicity conditions to all positive toughness values.

Provides a new sufficient condition linking toughness and degree sums for Hamiltonian cycles.

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. In this paper, we show 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.