Total chromatic number of generalized Mycielski graphs
Résumé
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 reveive 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 χt(G). The Total Coloring Conjecture (TCC) states that for every simple graph G, Δ(G)+1≤χt(G)≤Δ(G)+2. G is called Type 1 (resp. Type 2) if χt(G)=Δ(G)+1 (resp. χt(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) is Type 1.