On the clique number of the square of a line graph and its relation to maximum degree of the line graph

In 1985, Erdos and Nesetril conjectured that the square of the line graph of a graph G, that is, L(G)(2), can be colored with 5/4 Delta(G)(2) colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is, omega(L(G)(2)), is at most 5/4 Delta(G)(2). In 2015, Sleszynska-Nowak proved that omega(L(G)(2)) <= 3/2 Delta(G)(2). In this paper, we prove that omega(L(G)(2)) <= 4/3 Delta(G)(2). This theorem follows from our stronger result that omega(L(G)(2)) <=sigma G)(2)/3 where sigma(G). max(uv is an element of E(G)) d (u) + d (v) = Delta(L(G)) + 2.