NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2, ... , k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge is an element of E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By (ndi)(Sigma)(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) (ndi)(Sigma)( G) <= max{2 Delta(G) + 1, 25} if G is a planar graph, (2) (ndi)(Sigma)(G) <= max{2 Delta(G), 19} if G is a graph such that mad(G) <= 5.