AI 中文总结
研究图中圈长和弦相关的三个问题，包括(k + 1) - 临界图的连续圈长、k - 临界图的带弦奇圈及特定顶点度图的带弦圈问题，通过否定、反驳及证明等方式得出相应结论。
AI 中文摘要
我们主要考虑图中关于圈长和带弦圈的三个问题：(a) Gao、Huo和Ma提出，对于每个固定的k≥3，是否存在函数fk(n)→∞，使得每个n顶点(k + 1) - 临界图包含fk(n)个连续的圈长。(b) 设gk(n)是最大整数t，使得每个n顶点k - 临界图（k≥4）包含一个至少有t条弦的奇圈。Voss猜想gk(n)随n趋于无穷。(c) Kára和Král猜想每个31个顶点且最小度至少为8的图包含一个至少有31条弦的圈。我们对k = 3否定了问题(a)，对所有k≥5反驳了猜想(b)，证明了猜想(c)，并在结语部分讨论了另外两个相关问题。
英文摘要
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 Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). (c) Kára and Král \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 Erdős 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.
Comments18 pages
