On The Study Of D(d)-vertex-distinguishing Total Colorings Of Cycles

For a proper total coloring of a graph G=(V, E), the palette C(v) of a vertex u∈V is the set of the colors of the edges incident with v and the color of the vertex itself. Suppose u and v are two vertices of G. If C(u)≠C(v) then u and v are said to be distinguished by the total coloring. A d-strong total coloring (i.e.,D(d)-vertex distinguishing total coloring) of G is a proper total coloring that distinguishes all pairs of vertices u and v with distance 1≤dG(u,v)≤d. The d-strong total chromatic number (i.e., D(d)-vertex distinguishing total chromatic number) x_d~n(G) of G is the minimum number of colors of a d-strong total coloring of G.We determine x_d~n(Cn) completely when d ∈ [24,83] (d ∈ N) in Chapter 2. We will determine x_d~n(Cn) when d ∈ [(?), (?)-1] (d ∈ N, k≥ 4) and n≤ (?) in Chapter 3.