Total chromatic number of generalized Mycielski graphs

A total coloring of a simple graph GG is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. The minimum number of colors required for a proper total coloring of GG is called the total chromatic number of GG and denoted by χt(G)χt(G). The Total Coloring Conjecture (TCC) states that for every simple graph GG, Δ(G)+1≤χt(G)≤Δ(G)+2Δ(G)+1≤χt(G)≤Δ(G)+2. GG is called Type 1 (resp. Type 2) if χt(G)=Δ(G)+1χt(G)=Δ(G)+1 (resp. χt(G)=Δ(G)+2χt(G)=Δ(G)+2). In this paper, we prove that the generalized Mycielski graphs satisfy TCC. Furthermore, we get that if Δ(G)≤∣V(G)∣−12, then the generalized Mycielski graph μm(G)μm(G) is Type 1.