AI 中文总结
研究不含短奇洞的{帽，偶洞} - 自由图的着色问题，通过肯定回答相关问题，证明了对于整数\(q\geq3\)，此类图满足\(\chi(G)\leq \left\lceil\frac{2q + 1}{2q}\omega(G)\right\rceil\)且界是紧的。
AI 中文摘要
一个“洞”是长度至少为4的诱导圈，“偶洞”是长度为偶数的洞。“帽”是通过在洞上添加一个与洞上恰好两个连续顶点相邻的顶点得到的。Chen、Xu和Xu证明了每个{帽，偶洞} - 自由图\(G\)满足\(\chi(G)\leq \left\lceil\frac{5}{4}\omega(G)\right\rceil\)，当排除5 - 洞时将此界改进为\(\chi(G)\leq \left\lceil\frac{7}{6}\omega(G)\right\rceil\)。他们提出问题：对于每个整数\(q\geq3\)，每个不含长度至多为\(2q - 1\)的奇洞的{帽，偶洞} - 自由图\(G\)是否满足\(\chi(G)\leq \left\lceil\frac{2q + 1}{2q}\omega(G)\right\rceil\)。我们肯定地回答了这个问题，并表明对于每个\(q\geq3\)该界是紧的。
英文摘要
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 $χ(G)\leq \left\lceil\frac{5}{4}ω(G)\right\rceil$, and improved this bound to $χ(G)\leq \left\lceil\frac{7}{6}ω(G)\right\rceil$ 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 $$ χ(G)\leq \left\lceil\frac{2q+1}{2q}ω(G)\right\rceil. $$ We answer this question affirmatively and show that the bound is sharp for every $q\geq3$.
