Vertex Distinguishing IE-Total Coloring And General-Total Coloring Of K_5,n,p（5≤n≤8,n≤p）

The graph coloring is an important research direction in graph theory,and there are many research branches and rich research results.A general-total coloring of a graph G refers to a coloring method in which several colors are assigned to the vertices and edges of the graph G.A general-total coloring of a graph G is called an IE-total coloring if any two adjacent vertices of the graph G have different colors.Let h be a general-total coloring(resp.an IE-total coloring)of a graph G,for any a vertex y of the graph G,the(non-multiple)set composed of the color of y and the colors of all edges incident with y is called the color set of vertex y under h.which is denoted by Ch(y)or C(y).Let h be a general-total coloring(resp.an IE-total coloring)of a graph G,if for any two different vertices in the graph G and their color sets are different,then h is called a vertex distinguishing general-total coloring(resp.vertex distinguishing IE-total coloring)of the graph G,for short GVDTC(resp.VDIETC).The minimum number of colors that make a graph G has a GVDT coloring(resp.VDIET coloring)is called the vertex distinguishing generaltotal chromatic number(resp.vertex distinguishing IE-total chromatic number)of the graph G,which is denoted by χgvt(G)(resp.Xvtie(G)).In this paper,the vertex distinguishing IE-total colorings and vertex distinguishing general-total colorings of complete tripartite graphs K5,n,p(5≤n≤8,p≥n)are discussed by using of the methods of combinatorial analysis,constructing the method of distributing the color sets in advance for coloring.The vertex distinguishing IE-total chromatic numbers and vertex distinguishing general-total chromatic numbers of K5,n,p(5≤n≤8,p≥n)are determined in this paper.