The total chromatic number of split-indifference graphs

The total chromatic number of a graph G , χ T ( G ) , is the least number of colours sufficient to colour the vertices and edges of a graph such that no incident or adjacent elements (vertices or edges) receive the same colour. The Total Colouring Conjecture (TCC) states that every simple graph G has χ T ( G ) ≤ Δ ( G ) + 2 , and it is a challenging open problem in Graph Theory. For both split graphs and indifference graphs, the TCC holds, and χ T ( G ) = Δ ( G ) + 1 when Δ ( G ) is even. For a split-indifference graph G with odd Δ ( G ) , we give conditions for its total chromatic number to be Δ ( G ) + 2 , and we build a ( Δ ( G ) + 1 ) -total colouring otherwise. Also, we pose a conjecture for a class of graphs that generalizes split-indifference graphs.