Enhancing the Erd\H{o}s-Lov\'asz Tihany Conjecture for line graphs of multigraphs

TL;DR

This paper proves an enhanced version of the Erd ext{"o}s-Lovász Tihany Conjecture for line graphs of multigraphs, establishing new bounds for graph partitions based on chromatic and clique numbers.

Contribution

It introduces an improved theorem extending the conjecture for line graphs of multigraphs with additional parameters and bounds.

Findings

Proves the conjecture for line graphs of multigraphs with specific bounds.

Establishes partition results for graphs with chromatic number exceeding clique number.

Provides a generalized framework for graph coloring partitions.

Abstract

In this paper, we prove an enhanced version of the Erd\H{o}s-Lov\'asz Tihany Conjecture for line graphs of multigraphs. That is, for every graph $G$ whose chromatic number $\chi(G)$ is more than its clique number $\omega(G)$ and for nonnegative integer $\ell$, any two integers $s,t \geq 3.5\ell+2$ with $s+t = \chi(G)+1$, there is a partition $(S,T)$ of the vertex set $V(G)$ such that $\chi(G[S])\geq s$ and $\chi(G[T])\geq t+\ell$. In particular, when $\ell=1$, we can obtain the same result just for any $s,t\geq4$. The Erd\H{o}s-Lov\'asz Tihany conjecture is a special case when $\ell=0$.