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 $G$.