Abstract
The total chromatic number χ T ( 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, χ T ( 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.
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