A Note on Hamilton Cycles

TL;DR

This paper discusses conditions under which a graph with certain toughness and minimum degree properties guarantees the existence of a Hamilton cycle, contributing to graph theory's understanding of Hamiltonicity.

Contribution

It establishes that graphs with toughness greater than one and sufficiently high minimum degree are Hamiltonian, extending previous results in the field.

Findings

Graphs with toughness > 1 and high minimum degree are Hamiltonian.

The minimum degree condition is close to half the number of vertices.

Results hold for sufficiently large graphs.

Abstract

If $G$ is a more than one tough graph on $n$ vertices with $\delta\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.