Enhancing the Erdos-Lovasz Tihany Conjecture for line graphs of multigraphs

In this paper, we prove an enhanced version of the Erdos-Lovasz Tihany Conjecture for line graphs of multigraphs. That is, for every line graph G $G$ whose chromatic number chi(G) $\chi (G)$ is more than its clique number omega(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=chi(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 chi(G[S])>= s $\chi (G[S])\ge s$ and chi(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 Erdos-Lovasz Tihany conjecture for line graphs is a special case when l=0 $\ell =0$.