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An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 2(G) for trees, cacti, complete multipartite graphs and graph joins

M_2-edge colorings of cacti and graph joins