Enhancing the Erdős‐Lovász Tihany Conjecture for line graphs of multigraphs

In this paper, we prove an enhanced version of the Erdős‐Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every line graph G $G$ whose chromatic number χ(G) $\chi (G)$ is more than its clique number ω(G) $\omega (G)$ and for any nonnegative integer ℓ $\ell $, any two integers s,t≥ 3.5ℓ+ 2 $s,t\ge 3.5\ell +2$ with s+t=χ(G)+1 $s+t=\chi (G)+1$, there is a partition (S,T ) $(S,T)$ of the vertex set V(G) $V(G)$ such that χ(G[S])≥s $\chi (G[S])\ge s$ and χ(G[T])≥t+ℓ $\chi (G[T])\ge t+\ell $. In particular, when ℓ=1 $\ell =1$, we can obtain the same result just for any s,t≥4 $s,t\ge 4$. The Erdős‐Lovász Tihany conjecture for line graphs is a special case when ℓ=0 $\ell =0$.