On the Total-Neighbor-Distinguishing Index by Sums

We consider a proper coloring c of edges and vertices in a simple graph and the sum f(v) of colors of all the edges incident to v and the color of a vertex v. We say that a coloring c distinguishes adjacent vertices by sums, if every two adjacent vertices have different values of f. We conjecture that colors suffice to distinguish adjacent vertices in any simple graph. In this paper we show that this holds for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.