Clique number of the square of a line graph

We prove that the clique number of the square of a line graph of a graph GG is at most 1.5ΔG2 and that the fractional strong chromatic index of GG is at most 1.75ΔG2.An edge coloring   of a graph GG is strong if each color class is an induced matching of GG. The strong chromatic index   of GG, denoted by χs′(G), is the minimum number of colors for which GG has a strong edge coloring. The strong chromatic index of GG is equal to the chromatic number of the square of the line graph of GG. The chromatic number of the square of the line graph of GG is greater than or equal to the clique number of the square of the line graph of GG, denoted by ω(L)ω(L).In this note we prove that ω(L)≤1.5ΔG2 for every graph GG. Our result allows to calculate an upper bound on the fractional strong chromatic index of GG, denoted by χfs′(G). We prove that χfs′(G)≤1.75ΔG2 for every graph GG.