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

The total chromatic number ξT(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 is odd order and regular of degree d ⩾[37−1)6]|V(G)|, then a necessary and sufficient condition for ξT(G) = d + 1 is given. This improves a recent result of Chetwynd et al. (1991).