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

For a given graph G, denote by G"@D the subgraph of G induced by the vertices of maximum degree. The total chromatic number @g"T(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 K"2 is of even order, G"@D is a forest, and @d(G)+@D(G)>=32(|V(G)|-1) then @g"T(G)=@D(G)+1. We also show that for graphs G of even order and @d(G)+@D(G)>=32|V(G)|-52 we have that @g"T(G)=<@D(G)+2.