A bound on the total chromatic number (Q1297726)

A total colouring of a graph \(G\) is an assignment of colours to its vertices and edges so that no two adjacent edges have the same colour, no two adjacent vertices have the same colour, and no edge has the same colour as one of its endpoints. The total chromatic number \(\chi''(G)\) is the least number of colours required for a total colouring of \(G\). The authors prove that the total chromatic number of any graph with maximum degree \(\Delta\) is at most \(\Delta\) plus an absolute constant. In particular, they show that for \(\Delta\) sufficiently large, the total chromatic number of such a graph is at most \(\Delta+10^{26}\). The method is probabilistic.