Total Thue colourings of graphs

We show that the strong total Thue chromatic number \(\pi _\mathrm{T}(G) < 15\mathrm{\Delta }^2\) for a graph \(G\) with maximum degree \(\mathrm{\Delta }\ge 3\), and establish some other upper bounds for the weak and strong total Thue chromatic numbers 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. Moreover, considering the list version of the problem we show that the total Thue choice number of a graph is less than \(18\mathrm{\Delta }^2\).