A note on coloring (even-hole,cap)-free graphs

A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to exactly two adjacent vertices on the hole. In this note, we use a decomposition theorem by Conforti et al. (1999) to show that if a graph $G$ does not contain any even hole or cap as an induced subgraph, then $\chi(G)\le \lfloor\frac{3}{2}\omega(G)\rfloor$,where$\chi(G)$and$\omega(G)$ are the chromatic number and the clique number of $G$, respectively. This bound is attained by odd holes and the Hajos graph. The proof leads to a polynomial-time $3\mathrm{/}2$-approximation algorithm for coloring (even-hole,cap)-free graphs.