Some Cases of the Erd\H{o}s-Lov\'asz Tihany Conjecture for Claw-free Graphs

TL;DR

This paper proves the Erdős-Lovász Tihany Conjecture for certain pairs of parameters in claw-free graphs, expanding the classes of graphs where the conjecture is confirmed.

Contribution

The paper establishes the conjecture for pairs (s, t) with t ≤ s+2 when G contains a K_s, and for t ≤ 4s-3 when G is claw-free and contains a K_s, including the specific case (3, 10).

Findings

Confirmed the conjecture for t ≤ s+2 with K_s in G.

Proved the conjecture for t ≤ 4s-3 in claw-free graphs with K_s.

Validated the conjecture for the pair (3, 10) in claw-free graphs.

Abstract

The Erd\H{o}s-Lov\'asz Tihany Conjecture states that any $G$ with chromatic number $\chi(G) = s + t - 1 > \omega(G)$, 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 Erd\H{o}s Lov\'asz Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs.