Research On The Adjacent Vertex Distinguishing Proper Edge Colorings Of Several Classes Of Graphs

Let G(V, E) be a simple finite connect graph with order at least3, k be a positive integer and f be an edge coloring (i.e.,f be an assignment of k colors,1,2,…,k, to the edges of G). For each element z∈E(G), we use f(z) denote the color of z. For every vertex x∈V(G) the set of colors of edges incident with x is denoted by S(x) and is called the color set of x. If f is proper and S(u)≠S(v) for each edge uv∈E(G), then f is called a k-adjacent vertex distinguishing proper edge coloring of G (or a k-AVDPEC). 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 Xa(G). In the paper, the adjacent vertex distinguishing proper edge coloring of several classes of complete5-partite graphs,Pl□Pm□Pn, Pl□Cm□Pn and Cl□Cm□Pn are discussed by combinatorial analysis method, the adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete5-partite graphs, Pl□Pm□Pn, Pl□Cm□Pn and Cl□Cm□Pn are obtained.