Abstract: Let G be a simple graph. Suppose f is a general total coloring of graph G(i.e., an assignment of several colors to all vertices and edges of G), if any two adjacent vertices and any two adjacent edges of graph G are assigned different colors, then f is called an Ⅰ-total coloring of a graph G; if any two adjacent edges of G are assigned different colors, then f is called a Ⅵ-total coloring of a graph G. For an Ⅰ-total coloring(or Ⅵ-totalcoloring)f of a graph G, if C(u)≠C(v) for any two distinct vertices u and v of V(G), where C(x) denotes the set of colors of vertex x and the edges incident with x under f, then f is called a vertex distinguishing Ⅰ-total coloring(or vertex distinguishing Ⅵ-total coloring)of G. Let χⁱ_vt(G)=min{k|G has a k-VDIT coloring}, then χⁱ_vt(G) is called the VDIT chromatic number of G. Let χ^vi_vt(G)=min{k|G has a k-VDVIT coloring}, then χ^vi_vt(G) is called the VDVIT chromatic number of G. The VDIT coloring(or VDVIT coloring)of almost complete graphs and the VDIT chromatic number(VDVIT chromatic number)of them has been obtained by using analytical method and proof by contradiction.

Key words: complete graph, Ⅰ-total coloring, vertex-distinguishing Ⅰ-total coloring, vertex-distinguishing Ⅰ-total chromatic number