On a conjecture of Bondy and Vince

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 $k$.