An upper bound of star-edge-star total chromatic number of graphs

A concept of star-edge-star total coloring of graphs was presented.A proper total coloring of a graph G would be called star-edge-star total coloring if its vertices were star coloring and its edges were star-edge coloring.The star-edge-star total chromatic number of G was defined and denoted by χTss(G).The star-edge-star total chromatic number of some particular graphs(path,cycle,wheel,fan,complete graph) were given by using the method of coloring construction.Meantime,an upper bound of star-edge-star total chromatic number of graph which satisfied definite conditions,was given by means of probabilistic method,namely χTss(G)≤24(Δ-1)3/2 for a graph G if its maximum degree Δ(G)≥30.