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arXiv · 2607.27850

Optimal binding function for (cap,even hole)-free graphs with no short odd holes

Chenglong Deng·Xuding Zhu
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Abstract

A hole in a graph is an induced cycle of length at least $4$. A cap is a hole together with a vertex adjacent to exactly two consecutive vertices of it. Chen, Xu and Xu conjectured that if $q\ge2$ and $G$ is a $(\mathrm{cap},\mathrm{even\ hole})$-free graph with no odd hole of length at most $2q-1$, then $\chi(G)\le \left\lceil \frac{2q+1}{2q}\omega(G)\right\rceil.$ They confirmed the conjecture for $q \le 3$. In this paper, we prove the conjecture for all $q \ge 3$. As a corollary, we prove that for such a graph $G$, $\chi_f(G)\le \frac{2q+1}{2q}\omega(G).$

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Chenglong Deng, Xuding Zhu. 2026-07-30. Optimal binding function for (cap,even hole)-free graphs with no short odd holes. https://arxiv.org/abs/2607.27850

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