A result on the total colouring of powers of cycles

The total chromatic number χT (G) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that for every simple graph G, χT (G) ⩽ Δ(G)+2. This work verifies the TCC for powers of cycles C n k , n even and 2 < k < ⌊n/2⌋, showing that there exists and can be polynomially constructed a (Δ(G) + 2)-total colouring for these graphs.