Degree Conditions for Dominating Cycles in 1-tough Graphs

TL;DR

This paper establishes degree conditions under which longest cycles in 1-tough graphs are dominating, showing that high minimum degree ensures domination unless the graph belongs to a specific exceptional class.

Contribution

It proves new degree-based criteria for dominating cycles in 1-tough graphs, extending understanding of cycle structure under toughness and connectivity constraints.

Findings

Longest cycles are dominating in 1-tough graphs with high minimum degree, except for specific classes.

3-connected 1-tough graphs with certain degree conditions have dominating longest cycles.

Abstract

We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $\delta$ with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $\kappa(G)=2$ and $\tau(G)=1$. The second result follows immediately from the first result: (ii) if $G$ is a 3-connected 1-tough graph with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.