Total Chromatic Number of Graphs of High Maximum Degree

The total chromatic number \(\chi_T(G)\) 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 maximum degree \(\Delta(G) > \frac{3}{4} |V(G)I – \frac{1}{2} \), then \(\chi_T(G) \leq \Delta(G) + 2\). A slightly weaker version of the result has earlier been proved by Hilton and Hind \([9]\). The proof here is shorter and simpler than the one given in \([9]\).