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

Let G be a simple graph,k is a positive integer and f is a mapping form V(G)∪E(G) to {1,2,…,k},if for any uv∈E(G),there are f(u)≠f(v),f(u)≠f(uv),f(v)≠f(uv);for any uv,vw∈E(G),u≠w,there is f(uv)≠f(uw),then f is called a k-proper total coloring of G(k-TC in brief),and the number ΧT(G)=min{k|G存在k-TC} is called the total chromatic number of G.

Then the total chromatic number of G is Δ(G)+1 if(Δ,k)∈{(6,4),(5,5),(4,11)}.

The total chromatic number x1(G) is the smallest interger k such that G has a total k-coloring.

This paper has solved the open question proposed by Zhang Zhongfu and others in reference [1],which is to ascertain the attainable lower bound of x4T(G)+x4T(G) ,in which x4T(G) indicates the 4-total chromatic number of a graph G, while G indicates the complement graph of G.

The least number of k spanning over all star total-colorings of G,denoted by χst(G) is called the star total chromatic number of G.

The least number of k spanning over all star total k-colorings of G,denoted by χst(G),is called the star total chromatic number of G.

In this paper,we searched the star total chromatic numbers of Mycielski\'s graphs of complete bipartite graphs,complete graphs and complete multipartite graphs with equipotent parts and obtained the star total chromatic numbers.