Large cycles in 4-connected graphs

TL;DR

This paper investigates the structure of 4-connected graphs, establishing conditions under which they contain long cycles or have all longest cycles as dominating, thus advancing understanding of cycle properties in highly connected graphs.

Contribution

It provides a new characterization of 4-connected graphs relating minimum degree, connectivity, and the nature of their longest cycles.

Findings

Graphs either contain long cycles of length at least 4δ - κ - 4 or all longest cycles are dominating.

The result links minimum degree and connectivity to cycle length and dominance.

It extends previous results on cycle structure in highly connected graphs.

Abstract

Every 4-connected graph $G$ with minimum degree $\delta$ and connectivity $\kappa$ either contains a cycle of length at least $4\delta-\kappa-4$ or every longest cycle in $G$ is a dominating cycle.