Partition line graphs of multigraphs into two subgraphs with large chromatic numbers

Abstract

Wang and Yu (2022) prove an enhanced version of the Erdős–Lovász Tihany conjecture for line graphs of multigraphs. That is, let s,t and ℓ be arbitrary integers with t≥s≥3.5ℓ+2, ℓ≥0. If the line graph L(G) of some multigraph G has chromatic number s+t−1>ω(L(G)), then there is a partition (S,T) of the vertex set V(L(G)) such that χ(L(G)[S])≥s and χ(L(G)[T])≥t+ℓ. In this paper, for integers s and t with t≥s≥7, we prove that for each line graph L(G) with χ(L(G))=s+t−1≥ω(L(G)), there is a partition (S,T) of the vertex set V(L(G)) such that χ(L(G)[S])≥s and χ(L(G)[T])≥t+2, which slightly improves the recent result of Wang and Yu (2022) for ℓ=2.