Clique number of the square of a line graph

TL;DR

This paper establishes an upper bound on the clique number of the square of a line graph, which in turn provides bounds on the fractional strong chromatic index of any graph, advancing understanding of strong edge colorings.

Contribution

It proves a new upper bound on the clique number of the square of a line graph, enabling improved bounds on the fractional strong chromatic index.

Findings

Clique number of the square of a line graph is at most 1.5 times the square of maximum degree.

Upper bound for fractional strong chromatic index is 1.75 times the square of maximum degree.

Provides theoretical bounds relevant to strong edge coloring problems.

Abstract

An \emph{edge coloring} of a graph $G$ is strong if each color class is an induced matching of $G$. The \emph{strong chromatic index} of $G$, denoted by $\chi _{s}^{\prime }(G)$, is the minimum number of colors for which $G$ has a strong edge coloring. The strong chromatic index of $G$ is equal to the chromatic number of the square of the line graph of $G$. The chromatic number of the square of the line graph of $G$ is greater than or equal to the clique number of the square of the line graph of $G$, denoted by $\omega(L)$. In this note we prove that $\omega(L) \le 1.5 \Delta_{G}^2$ for every graph $G$. Our result allows to calculate an upper bound for the fractional strong chromatic index of $G$, denoted by $\chi_{fs}^\prime(G)$. We prove that $\chi_{fs}^{\prime}(G) \le 1.75 \Delta_G^2$ for every graph $G$.