Let G = (V ; E) be a simple undirected graph without loops and multiple edges, the vertex and edge sets of it are represented by V = V (G) and E = E(G), respectively. A weighting w of the edges of a graph G induces a coloring of the vertices of G where the color of vertex v, denoted $S_v:={Sigma}_{e{ i}v};w(e)$. A k-edge-weighting of a graph G is an assignment of an integer weight, w(e) ${in}${1,2,...,k} to each edge e, such that two vertex-color $S_v$, $S_u$ be distinct for every edge uv. In this paper we determine an exact 3-edge-weighting of complete graphs $k_{3q+1};{forall}_q;{in};{mathbb{N}}$. Several open questions are also included.