Colouring squares of claw-free graphs

• Published in: Electronic notes in discrete mathematics Vol. 61; pp. 663 - 669
• Main Authors: de Verclos, Rémi de Joannis, Kang, Ross J., Pastor, Lucas
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2017
• Subjects: claw-free graphs; Erdős–Nešetřil conjecture; Graph colouring; Erdős–Nešetřil conjecture; claw-free graphs; Graph colouring
• ISSN: 1571-0653; 1571-0653
• DOI: 10.1016/j.endm.2017.07.021

Is there some absolute ε>0 such that for any claw-free graph G, the chromatic number of the square of G satisfies χ(G2)≤(2−ε)ω(G)2, where ω(G) is the clique number of G? Erdős and Nešetřil asked this question for the specific case of G the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil.