  A well-studied concept is that of the total chromatic number. A proper total
colouring of a graph is a colouring of both vertices and edges so that every
pair of adjacent vertices receive different colours, every pair of adjacent
edges receive different colours and every vertex and incident edge receive
different colours. This paper considers a strengthening of this condition and
examines the minimum number of colours required for a total colouring with the
additional property that for any adjacent vertices $u$ and $v$, the sets of
colours incident to $u$ is different from the set of colours incident to $v$.
It is shown that there is a constant $C$ so that for any graph $G$, there
exists such a colouring using at most $\Delta(G) + C$ colours.
