An existence criterion for a cycle such that the vertex set beyond this cycle is independent

TL;DR

This paper establishes a precise condition on 2-connected graphs ensuring the existence of a cycle whose removal leaves an independent vertex set, advancing understanding of cycle structures in graph theory.

Contribution

It provides a sharp threshold for minimum degree guaranteeing a cycle with an independent vertex set outside it in 2-connected graphs.

Findings

Proves the existence of such a cycle under the given degree condition.

Shows the result is optimal and cannot be improved.

Enhances understanding of cycle and independence properties in graphs.

Abstract

We prove that if $G$ is a 2-connected graph with $\delta(G) \geqslant \frac{v(G) + 2}{3}$ then $G$ has a cycle $W$ such that $V(G - W)$ is independent. This result is best possible in the sense that it becomes false if $\frac{v(G) + 2}{3}$ is replaced by any smaller number.