Research On Vertex Strongly Distinguishing Total Coloring Of Graphs

Let G=(V, E) be a simple graph, f be a mapping from V∪E to {1,2,…, k}, k be a positive integer. For each x∈V, let C(x)={f(x)}∪{f(y)|y∈V, y is adjacent to x}∪{f(e)|e∈E, e is incident with x}, C(x) is called the color set of x under f. If (i)(?) uv∈E, we have f (u)≠f(v), f(u)≠f (uv), f (v)≠(uv);(ii)(?) uv, uw∈E, v≠w, we have f (uv)≠f (uw), then f is called a proper total coloring of G using k colors, or k-PTC. For a k-PTC f, if (?) u, v∈V, u≠v, we have C(u)≠C(v), then f is called a vertex strongly distinguishing total coloring of G using k colors, or k-VSDTC of G is brief. The number xust(G)=min{k|G has a k-VSDTC} is called the vertex strongly distinguishing total chromatic number of G. In the paper, the vertex strongly distinguishing total coloring of star, fan, wheel, double star, complete bipartite graphs, complete graphs, cycle and path are discussed by combinatorial analysis method, the vertex strongly distinguishing total chromatic numbers of star, fan, wheel, double star and several classes of complete bipartite graphs, complete graphs, cycle and path are obtained.