On the vertex-distinguishing proper total colorings of several classes of complete p-partite graphs with equipotent parts

In this paper, we discuss the vertex-distinguishing proper total colorings of several classes of complete p-partite graphs with equipotent parts. We denote by K(p × q) the complete p-partite graph with cardinality q of each part. By constructing coloring in terms of the symmetry of regular polygons and the methods of combinatorial analysis, we will obtain vertex-distinguishing proper total chromatic numbers for K(p × q) when q = 2, 3, 4, 5, respectively.