# On a conjecture of Bondy and Vince

## TL;DR

This paper confirms Bondy and Vince's long-standing conjecture that graphs with a bounded number of low-degree vertices typically contain two cycles with lengths differing by one or two, except for finitely many counterexamples.

## Contribution

The paper proves the conjecture for all nonnegative integers k, extending previous partial results and resolving a 20-year-old open problem.

## Key findings

- Confirmed the conjecture for all k
- Identified only finitely many counterexamples
- Extended previous partial results

## Abstract

Twenty years ago Bondy and Vince conjectured that for any nonnegative integer $k$, except finitely many counterexamples, every graph with $k$ vertices of degree less than three contains two cycles whose lengths differ by one or two. The case $k\leq 2$ was proved by Bondy and Vince, which resolved an earlier conjecture of Erd\H{o}s et. al.. In this paper we confirm this conjecture for all $k$.