Vertex-Distinguishing IE-total Coloring And General-total Coloring Of K_1,n,p（1≤n≤6,n≤p）

Graph coloring is one of the most important topics in graph theory research.Let G be a simple graph.A general total coloring f of G is defined as follows:V(G)UE(G)→[1,k].Once uv ∈ E(G),there is f(u)≠f(v),then call f is IE-total coloring.For a k-IE—total coloring f of G[a general total coloring f of G]and any vertex x of G,let Cf(x)or C(x)denote the set of colors of vertex x and of the edges incident with x,that is C(x)={f(xu)|xu ∈ E} ∪{f(x)}.We call C(x)the color set of x.If C(u)≠C(v)for any two different vertices u and v of V(G),then f is called a vertex-distinguishing IE-total coloring(k-VDIETC)general vertex-distinguishing total coloring of G.Denote Xvtie(G)＝min{k|G has k-VDIETC}.Which is called the vertex-distinguishing IE-total chromatic number of G,χgvt(G)=min{k| G has general vertex-distinguishing total coloringlis called the vertex-distinguishing general total chromatic number of G.the vertex-distinguishing general total coloring of G is also called general vertex-distinguishing total coloring.so we abbreviate it to GVDTC.For each graph G,we use ni=n.i(G)express the number of vertex which degree is i,δ(G)≤ i≤△(G),ζ(G)meet demand(1l)+(2l)+(3l)+…+(s+1l)>≥nδ+nδ+1＋… +ns,δ≤s≤△.The minium positive integer set up for every number.we will probe the question on vertex-distinguishing IE-total colorings and general vertex-distinguishing total colorings of K1,n,p(1≤n≤6,n ≤p)by using the method of proof by contradiction、combinatorial analysis and constructing the concrete colorings.The vertex-distinguishing IE一total colorings and general vertex-distinguishing total colorings of K1,n,p(1≤n≤6,n≤p)are determined in this paper.The conclusions of the paper shows that,for this graph,χvtie(K1,n,p)[χgzvt(K1,p,p)]=ζ(K1,n,p)+1 orζ(K1,n,p).