Let G be an undirected graph without multiple edges and loops, V(G) be the set of vertices of the graph G, E(G) be the set of edges of the graph G. Denote by Kn;Km;n;Kl;m;n, respectively, a complete graph with n vertices, a complete bipartite graph with m vertices in one partition and with n vertices in another, a complete tripartite graph with l vertices in one partition, m vertices in the other part, and n vertices in the third partition. Terminologies and notations not defined here can be found in [6]. A proper edge coloring f of a graph G is called vertex-distinguishing if for any different vertices u; v 2 V (G); S(u; f) 6= S(v; f): The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by Â0 vd(G): The definition of vertex-distinguishing edge coloring of a graph was introduced in [1,2] and, independently, as the “observability” of a graph in [3-5]. In this work we obtain some results on vertex-distinguishing edge colorings of complete 3- and 4-partite graphs. In particular, the following results hold. Theorem 1. Let l,m and n be any natural numbers. Then formula