Enhancing the Erd\H{o}s-Lov\'asz Tihany Conjecture for graphs with independence number two

TL;DR

This paper proves an improved version of the Erdős-Lovász Tihany Conjecture for graphs with independence number two, showing such graphs are -splittable under certain conditions, with the result being optimal.

Contribution

The paper establishes a stronger form of the Erdős-Lovász Tihany Conjecture specifically for graphs with independence number two, extending known results.

Findings

Graphs with independence number two satisfy the enhanced conjecture.

The result is proven to be optimal with existing examples.

The conjecture holds for graphs where =+1.

Abstract

Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi(G[S])\geq s$ and $\chi(G[T])\geq t$. The well-known Erd\H{o}s-Lov\'asz Tihany Conjecture from 1968 states that every graph $G$ whose chromatic number $\chi(G)=s+t-1$ is more than its clique number $\omega(G)$ is $(s,t)$-splittable. In this paper, we prove an enhanced version of the Erd\H{o}s-Lov\'asz Tihany Conjecture for graphs with independence number two. That is, for every graph $G$ with $\chi(G)=s+t-1>\omega(G)+1$ is $(s,t+1)$-splittable. There are examples showing that this result is best possible.