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A well-studied concept is that of the total chromatic number. A proper totalcolouring of a graph is a colouring of both vertices and edges so that everypair of adjacent vertices receive different colours, every pair of adjacentedges receive different colours and every vertex and incident edge receivedifferent colours. This paper considers a strengthening of this condition andexamines the minimum number of colours required for a total colouring with theadditional property that for any adjacent vertices $u$ and $v$, the sets ofcolours 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 $G$, thereexists such a colouring using at most $\Delta(G) + C$ colours.

The adjacent vertex distinguishing total chromatic number