Halin-图的点强全染色(英文)

On the Vertex Strong Total Coloring of Halin-Graphs

图G(V,E)的一个k-正常全染色f叫做一个k-点强全染色当且仅当对任意v∈V(G), N[v]中的元素被染不同色,其中N[v]={u|uv∈V(G)}∪{v}．χTvs(G)=min{k|存在图G的k- 点强全染色}叫做图G的点强全色数．对3-连通平面图G(V,E),如果删去面fo边界上的所有点后的图为一个树图,则G(V,E)叫做一个Halin-图．本文确定了最大度不小于6的Halin- 图和一些特殊图的的点强全色数XTvs(G),并提出了如下猜想:设G(V,E)为每一连通分支的阶不小于6的图,则χTvs(G)≤△(G)+2,其中△(G)为图G(V,E)的最大度．

A proper k-total coloring f of the graph G（V, E） is said to be a k-vertex strong total coloring if and only if for every v ∈ V（G）, the elements in N[v] are colored with different colors, where N[v] =. {u｜uv E V（G）} ∪{v}. The value xT^vs（G） = min{k｜ there is a k-vertex strong total coloring of G} is called the vertex strong total chromatic number of G. For a 3-connected plane graph G（V, E）, if the graph obtained from G（V, E） by deleting all the edges on the boundary of a face f0 is a tree, then G（V, E） is called a Halin-graph. In this paper, xT^vs,8（G） of the Halin-graph G（V,E） with A（G） 〉 6 and some special graphs are obtained. Furthermore, a conjecture is initialized as follows： Let G（V, E） be a graph with the order of each component are at least 6, then xT^vs（G） ≤ △（G） ＋ 2, where A（G） is the maximum degree of G.