Clique number of the square of a line graph

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 $G$.

9 pages, 3 figures