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arXiv · 2607.15501

Cycle lengths and chords under chromatic and degree constraints

Xiaozheng Chen·Bo Ning
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Abstract

We mainly consider three problems on cycle lengths and cycles with chords in graphs: (a) Gao, Huo, and Ma \cite[Question~1.5]{GaoHuoMa2021} asked whether, for every fixed $k\ge3$, there is a function $f_k(n)\to\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. (b) Let $g_k(n)$ be the maximum integer $t$ such that every $n$-vertex $k$-critical graph with $k\ge4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to\infty$ as $n\to\infty$ for each $k\ge4$, which extends a 1976 conjecture of Erd\H{o}s (see also Erd\H{o}s Problem~1091 \cite{Bloom1091}). (c) K\'ara and Kr\'al \cite{KaraKral2003} conjectured that every graph on $31$ vertices with minimum degree at least $8$ contains a cycle with at least $31$ chords. We answer question (a) in the negative for $k=3$, and disprove conjecture (b) for all $k\ge5$. We point out the work of Alexeev-Putterman-Sawhney-Sellke-Valiant (2026) on Erd\H{o}s Problem 1901 disproves the case $k=4$ for conjecture (b). We prove conjecture (c). We also discuss two other related problems in the part of concluding remark.

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Xiaozheng Chen, Bo Ning. 2026-07-16. Cycle lengths and chords under chromatic and degree constraints. https://arxiv.org/abs/2607.15501

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