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 $\frac{5}{4}Δ(G)^2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $ω(L(G)^2)$, is at most $\frac{5}{4}Δ(G)^2$. In 2015, Śleszyńska-Nowak proved that $ω(L(G)^2)\le \frac{3}{2}Δ(G)^2$. In this paper, we prove that $ω(L(G)^2)\le \frac{4}{3}Δ(G)^2$. This theorem follows from our stronger result that $ω(L(G)^2)\le \frac{σ(G)^2}{3}$ where $σ(G) := \max_{uv\in E(G)} d(u) + d(v)$, is the Ore-degree of the graph $G$.

11 pages