The optimal binding function for (cap, even hole)-free graphs

A {\em hole} is an induced cycle of length at least 4, an {\em even hole} is a hole of even length, and a {\em cap} is a graph obtained from a hole by adding an additional vertex which is adjacent exactly to two adjacent vertices of the hole. A graph $G$ obtained from a graph $H$ by blowing up all the vertices into cliques is said to be a clique blowup of $H$. Let $p, q$ be two positive integers with $p>2q$, let $F$ be a triangle-free graph, and let $G'$ be a clique blowup of $F$ with $\omega(G')\leq\max\{\frac{2q(p-q-2)}{p-2q}, 2q\}$. In this paper, we prove that for any clique blowup $G$ of $F$, $\chi(G)\leq\lceil\frac{p}{2q}\omega(G)\rceil$ if and only if $\chi(G')\leq\lceil\frac{p}{2q}\omega(G')\rceil$. As its consequences, we show that every (cap, even hole)-free graph $G$ satisfies $\chi(G)\leq\lceil\frac{5}{4}\omega(G)\rceil$, which affirmatively answers a question of Cameron {\em et al.} \cite{CdHV2018}, we also show that every (cap, even hole, 5-hole)-free graph $G$ satisfies $\chi(G)\leq\lceil\frac{7}{6}\omega(G)\rceil$, and the bound is reachable.