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

Abstract: In 1985, Erd\H{o}s and Ne\v{s}et\v{r}il conjectured that the square of the line graph of a graph $G$, that is $L(G)<sup>2$, can be colored with $\frac{5}{4}\Delta(G)<sup>2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $\omega(L(G)<sup>2)$, is at most $\frac{5}{4}\Delta(G)<sup>2$. In 2015, \'Sleszy\'nska-Nowak proved that $$\omega(L(G)^2)\le \frac{3}{2}\Delta(G)<sup>2$$. In this paper, we prove that $$\omega(L(G)^2)\le \frac{4}{3}\Delta(G)<sup>2$$. This theorem follows from our stronger result that $$\omega(L(G)^2)\le \frac{\sigma(G)<sup>2}{3}$$ where $\sigma(G) := \max_{uv\in E(G)} d(u) + d(v)$, is the Ore-degree of the graph $G$.