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

TL;DR

This paper improves bounds on the clique number of the square of a line graph, relating it to the Ore-degree of the original graph, advancing understanding of graph coloring conjectures.

Contribution

It establishes a new upper bound on the clique number of the square of a line graph in terms of the Ore-degree, strengthening previous results.

Findings

Proves 4a1; 4a1; 4a1; bound 4a1; 4a1; rac{4}{3}\

Improves previous bounds from rac{3}{2}\

Relates clique number to Ore-degree for better bounds.

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 $L(G)^2$, 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, \'Sleszy\'nska-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 $G$.