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arXiv · 2607.25396

Optimal coloring of $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graphs with no short odd holes

Feng Liu·Shuang Sun·Yan Wang
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Abstract

A \emph{hole} is an induced cycle of length at least four, and an \emph{even hole} is a hole of even length. A \emph{cap} is obtained from a hole by adding a vertex adjacent to exactly two consecutive vertices of the hole. Chen, Xu, and Xu proved that every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ satisfies $\chi(G)\leq \left\lceil\frac{5}{4}\omega(G)\right\rceil$, and improved this bound to $\chi(G)\leq \left\lceil\frac{7}{6}\omega(G)\right\rceil$ when $5$-holes are also excluded. They asked whether, for every integer $q\geq3$, every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ with no odd hole of length at most $2q-1$ satisfies $$ \chi(G)\leq \left\lceil\frac{2q+1}{2q}\omega(G)\right\rceil. $$ We answer this question affirmatively and show that the bound is sharp for every $q\geq3$.

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Feng Liu, Shuang Sun, Yan Wang. 2026-07-28. Optimal coloring of $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graphs with no short odd holes. https://arxiv.org/abs/2607.25396

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