AI 中文总结
本文证明Chen等人针对无短奇孔的（帽，偶孔）-自由图提出的染色数猜想，将其验证范围从q≤3拓展至所有q≥3，并得到分数色数的相关推论。
AI 中文摘要
图中的孔指长度至少为4的诱导环，帽是一个孔加上仅与该孔中两个连续顶点相邻的顶点。Chen、Xu和Xu猜想，若q≥2，且G是不含长度至多2q-1的奇孔的（帽，偶孔）-自由图，则χ(G)≤⌈(2q+1)/(2q)ω(G)⌉。他们已验证q≤3时猜想成立，本文证明所有q≥3时该猜想成立，作为推论还得到此类图的分数色数χ_f(G)≤(2q+1)/(2q)ω(G)。
英文摘要
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 $χ(G)\le \left\lceil \frac{2q+1}{2q}ω(G)\right\rceil.$ 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 $G$, $χ_f(G)\le \frac{2q+1}{2q}ω(G).$
