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

TL;DR

This paper proves a chromatic bound for (even-hole, cap)-free graphs using a decomposition theorem and provides a polynomial-time approximation algorithm for their coloring.

Contribution

It establishes a new chromatic bound for (even-hole, cap)-free graphs and introduces a polynomial-time 3/2-approximation coloring algorithm based on this bound.

Findings

Chromatic number bound: χ(G) ≤ ⌊3/2 ω(G)⌋ for (even-hole, cap)-free graphs.

The bound is tight, achieved by odd holes and the Hajos graph.

A polynomial-time 3/2-approximation algorithm for coloring these graphs.

Abstract

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 $3/2$-approximation algorithm for coloring (even-hole,cap)-free graphs.