Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs

Sean Longbrake; Juvaria Tariq

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arXiv · 2406.15164

Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs

Sean Longbrake·Juvaria Tariq

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Abstract

The Erdős-Lovász Tihany Conjecture states that any $G$ with chromatic number $χ(G) = s + t - 1 > ω(G)$, with $s,t \geq 2$ can be split into two vertex-disjoint subgraphs of chromatic number $s, t$ respectively. We prove this conjecture for pairs $(s, t)$ if $t \leq s + 2$, whenever $G$ has a $K_s$, and for pairs $(s, t)$ if $t \leq 4 s - 3$, whenever $G$ contains a $K_s$ and is claw-free. We also prove the Erdős Lovász Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs.

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BibTeX RIS

Sean Longbrake, Juvaria Tariq. 2024-07-04. Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs. https://arxiv.org/abs/2406.15164

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