  A total colouring of a graph is a colouring of its vertices and edges such
that no two adjacent vertices or edges have the same colour and moreover, no
edge coloured $c$ has its endvertex coloured $c$ too. A weak total Thue
colouring of a graph $G$ is a colouring of its vertices and edges such that the
colour sequence of consecutive vertices and edges of every path of $G$ is
nonrepetitive. In a total Thue colouring also the induced vertex-colouring and
edge-colouring of $G$ are nonrepetitive. The weak total Thue number
$\pi_{T_w}(G)$ of a graph $G$ denotes the minimum number of colours required in
every weak total Thue colouring and the minimum number of colours required in
every total Thue colouring is called the total Thue number $\pi_T$. Here we
show some upper bounds for both parameters depending on the maximum degree or
size of the graph. We also give some lower bounds and some better upper bounds
for these graph parameters considering special families of graphs.
