A k-proper total coloring of graph G is called the star total coloring of G,if the Coloring of the vertices and edges of any path of length 2 in G are all different,the least number of k spanning over all k-star-total-colorings of G is called the star total chromatic number of G.

This paper defined the star-edge coloring and star chromatic index x′_s(G),and proved that if G is a graph withΔ≥7 then x′_s(G)≤[16(Δ-1)3/2].

Furthermore,a simple and feasible star-edge coloring method is also put forward.

introduced the concept of star-edge coloring of graphs, a star-edge coloring of graph G is a proper edge coloring of G such that any path of length 4 in G is not bicolored.

This paper discusses the properties of the critical graphs about the star colorings, and obstain several necessary conditions and sufficent conditions how to judge that a graph is critical graph about the star coloring.

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.