Total chromatic number of generalized Mycielski graphs

A total coloring of a simple graph G 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 G is called the total chromatic number of G and denoted by χ t ( G ) . The Total Coloring Conjecture (TCC) states that for every simple graph G , Δ ( G ) + 1 ≤ χ t ( G ) ≤ Δ ( G ) + 2 . G is called Type 1 (resp. Type 2) if χ t ( G ) = Δ ( G ) + 1 (resp. χ t ( 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 ) ∣ − 1 2 , then the generalized Mycielski graph μ m ( G ) is Type 1.