Erd\H{o}s-Lov\'asz Tihany Conjecture for graphs with forbidden holes

TL;DR

This paper provides new evidence supporting the Erdős-Lovász Tihany Conjecture by proving it holds for graphs with certain forbidden holes and independence number at least 3.

Contribution

It extends the conjecture's validity to graphs with no holes of specific lengths and independence number at least 3, a case previously unverified.

Findings

Graphs with no holes of length between 4 and 2α(G)-1 satisfy the conjecture.

The result applies to graphs with independence number at least 3.

Supports the conjecture for broader classes of graphs.

Abstract

A hole in a graph is an induced cycle of length at least $4$. Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi(G[S ]) \ge s$ and $\chi(G[T ]) \ge t$. The well-known Erd\H{o}s-Lov\'asz Tihany Conjecture from 1968 states that every graph $G$ with $\omega(G) < \chi(G) = s + t - 1$ is $(s,t)$-splittable. This conjecture is hard, and few related results are known. However, it has been verified to be true for line graphs, quasi-line graphs, and graphs with independence number $2$. In this paper, we establish more evidence for the Erd\H{o}s-Lov\'asz Tihany Conjecture by showing that every graph $G$ with $\alpha(G)\ge3$, $\omega(G) < \chi(G) = s + t - 1$, and no hole of length between $4$ and $2\alpha(G)-1$ is $(s,t)$-splittable, where $\alpha(G)$ denotes the independence number of a graph $G$.