Clique number of the square of a line graph

Abstract

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