The perfect divisibility and chromatic number of some odd hole-free graphs

Abstract

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. It is NP-hard to color the vertices of an odd hole-free graph. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ with at least one edge admits a partition of $V(H)$ into sets $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. $G$ is short-holed if every hole in $G$ has length 4. A hammer is the graph obtained by identifying one vertex of a $K_3$ and one end vertex of a $P_3$. In this paper, we prove that (i) (odd hole, hammer, $K_{2,3}$)-free graphs are perfectly divisible, (ii) $\chi(G)\le 4\omega(G)(\omega(G)-1)$ if $G$ is short-holed and $(K_1+C_4)$-free, (iii) $\chi(G)\le 2\omega(G)-1$ if $G$ is short-holed and $(K_1\cup K_3)$-free, and (iv) $\chi(G)\le 16\omega(G)-24$ if $G$ is short-holed and $(K_1+(K_1\cup K_3))$-free.