The total chromatic number of regular graphs of even order and high degree

The total chromatic number χT(G) of a graph G is the minimum number of colours needed to colour the edges and the vertices of G so that incident or adjacent elements have distinct colours. We show that if G is a regular graph of even order and δ(G)⩾23|V(G)|+236, thenχT(G)⩽Δ(G)+2.