Cycle lengths and chords under chromatic and degree constraints

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 $k\ge 3$, there is a function $f_k(n)\to \mathrm{\infty }$ 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 $k\ge 4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ for each $k\ge 4$, which extends a 1976 conjecture of Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). © 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 $k\ge 5$. 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 ©. We also discuss two other related problems in the part of concluding remark.