Total chromatic number of graphs of odd order and high degree Original Research Article

The total chromatic number χT(G) of a graph G is the least number of colours needed to colour the edges and vertices of G so that no incident or adjacent elements receive the same colour. This paper shows that if G has odd order, minimum degree δ(G) and r(G) vertices of maximum degree Δ(G) such that δ(G)+Δ(G)⩾32|V(G)|+r(G)+52, then χT(G)=Δ(G)+1.