Strong edge colorings of graphs Original Research Article

Let χs′(G), called the strong coloring number of G, denote the minimum number of colors for which there is a proper edge coloring of a graph G in which no two of its vertices is incident to edges colored with the same set of colors. It is shown that χs′(G) ⩽ ⌈cn⌉, 12 < c ⩽ 1, whenever Λ (G) is appropriately bounded as a function of n, where n is the order of G. This result is in the direction of the conjecture that χs′(G) ⩽ n + 1 for each graph G with no isolated edges and at most one isolated vertex.