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A proper k-edge coloring of a graph G is an assignment of k colors, 1, 2, · · · , k, to edges of G. For a proper edge coloring f of G and any vertex x of G, we use S(x) denote the set of the colors assigned to the edges incident to x. If for any two adjacent vertices u and v of G, we have S(u) = S(v), then f is called the adjacent vertex distinguishing proper edge coloring of G (or AVDPEC of G in brief). The minimum number of colors required in an AVDPEC of G is called the adjacent vertex distinguishing proper edge chromatic number of G, denoted by χ ′ a(G). In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 4-partite and 5-partite graphs are obtained. (Abstract) Keywords-complete 4-partite graphs; complete 5-partite graphs; proper edge coloring; adjacent vertex-distinguishing proper edge coloring (key words)

On the adjacent vertex distinguishing proper edge colorings of several classes of complete 4-partite and 5-partite graphs