The vertex strong total chromatic number of general graphs K(n,m);

A proper k-total coloring σ of graph G(V,E) is called a k-vertex strong total coloring of G(V,E) if and only if for v∈V(G),the elements in N[v] are colored with different colors,where N[v]={u|vu∈E(G)}∪{v};and χvsT(G)=min{k|there is a k-vertex strong total coloring of G} is called the vertex strong total chromatic number of G.

A proper k-total coloring σ of graph G(V,E)is called a k-vertex strong total coloring of G(V,E)if and only if for ν∈V(G),the elements in N[ν]are colored with different colors,where N[ν]={u|νu∈E(G)}∪{ν};and χ~(νs)_(_T)(G)=min{k|there is a k-vertex strong total coloring of G}is called the vertex strong total chromatic number of G.

In this paper, the edge-face total chromatic number of Halin graphs are obtained for Δ(H_g)≥7 and Δ(H_g) = 4,5,6 , where Δ(H_g) denote maximum degree of Halin graph H_g.

In this paper we give the best upper and lower bounds of the edge face chromatic number χ ef for all 1-trees.