A Degree Condition for Dominating Cycles in $t$-tough Graphs with $t>1$

TL;DR

This paper proves that in t-tough graphs with t>1, a certain minimum degree condition guarantees that all longest cycles are dominating, extending understanding of cycle structure in tough graphs.

Contribution

It establishes a new degree condition ensuring that longest cycles in t-tough graphs are dominating, advancing cycle theory in graph toughness.

Findings

Longest cycles are dominating under the degree condition

The degree bound is (n-2)/3

Applicable for graphs with t>1

Abstract

Let $G$ be a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$. It is proved that if $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.