On the choosability of claw-free perfect graphs

TL;DR

This paper proves that for claw-free perfect graphs with maximum clique size at most 4, the choice number equals the chromatic number, supporting a conjecture about their colorability.

Contribution

It establishes the conjecture for a significant subclass of claw-free perfect graphs using a decomposition approach.

Findings

Conjecture holds for claw-free perfect graphs with maximum clique size ≤ 4.

Decomposition into elementary and peculiar graphs is key to the proof.

Supports broader conjecture about claw-free graphs' choosability.

Abstract

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most $4$.