On the chromatic number of a family of odd hole free graphs

TL;DR

This paper extends known results on coloring (odd hole, $K_4$)-free graphs to a broader class, showing they are nearly perfect with chromatic number at most one more than their clique number, with specific conditions for equality.

Contribution

It generalizes previous coloring bounds to (odd hole, full house)-free graphs and characterizes when the bound is tight.

Findings

For (odd hole, full house)-free graphs, χ(G) ≤ ω(G) + 1.

Equality holds iff ω(G) = 3 and G contains H as an induced subgraph.

The proof technique extends prior methods used for (odd hole, $K_4$)-free graphs.

Abstract

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not has $H$ as an induced subgraph. In this paper, we use the proving technique of Chudnovsky et al to generalize this conclusion to (odd hole, full house)-free graphs, and prove that for (odd hole, full house)-free graph $G$, $\chi(G)\le \omega(G)+1$, and the equality holds if and only if $\omega(G)=3$ and $G$ has $H$ as an induced subgraph.