On the clique number of the square of a line graph and its relation to Ore-degree

Abstract

In 1985, Erdős and Nešetřil conjectured that the square of the line graph of a graph $G$, that is $L(G{)}^2$, can be colored with $\frac{5}{4}\Delta (G{)}^2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $\omega (L(G{)}^2)$, is at most $\frac{5}{4}\Delta (G{)}^2$. In 2015, Śleszyńska-Nowak proved that $\omega (L(G{)}^2)\le \frac{3}{2}\Delta (G{)}^2$. In this paper, we prove that $\omega (L(G{)}^2)\le \frac{4}{3}\Delta (G{)}^2$. This theorem follows from our stronger result that $\omega (L(G{)}^2)\le \frac{\sigma (G{)}^2}{3}$ where $\sigma (G):={\max }_{uv\in E(G)}d(u)+d(v)$, is the Ore-degree of the graph $G$.

11 pages