On the Erd\H{o}s-Gy\'arf\'as conjecture in claw-free graphs

TL;DR

This paper investigates the Erdős-Gyárfás conjecture within claw-free graphs, especially focusing on cubic claw-free graphs, to determine if such graphs necessarily contain cycles of length a power of two.

Contribution

The paper provides new results on the Erdős-Gyárfás conjecture specifically for claw-free graphs, including cubic cases, advancing understanding of the conjecture in this graph class.

Findings

Results on the conjecture for claw-free graphs

Specific findings for cubic claw-free graphs

Progress towards the conjecture in restricted graph classes

Abstract

The Erd\H{o}s-Gy\'{a}rf\'{a}s conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erd\H{o}s-Gy\'{a}rf\'{a}s conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.