A result on the total colouring of powers of cycles ?
The total chromatic number ?T(G) ? 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 , ?T(G)??(G)+2 ? T ( G ) ? ? ( G ) + 2 . This work verifies the TCC for powers of cycles C n k , n even and 2<k<n/2 2 < k < n / 2 , showing that there exists and can be polynomially constructed a (?(G)+2) ( ? ( G ) + 2 ) -total colouring for these graphs.
total colouring powers of cycles total chromatic number total colouring conjecture