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We show that the strong total Thue chromatic number π T (G) < 15 2 for a graph G with maximum degree ≥ 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 2 .

Let G be a plane graph. A vertex-colouring ϕ of G is called facial non-repetitive if for no sequence r 1 r 2 . . . r 2n , n ≥ 1, of consecutive vertex colours of any facial path it holds r i = r n+i for all i = 1, 2, . . . , n. A plane graph G is facial non-repetitively l-choosable if for every list assignment L : V → 2 N with minimum list size at least l there is a facial non-repetitive vertex-colouring ϕ with colours from the associated lists. The facial Thue choice number, π f l (G), of a plane graph G is the minimum number l such that G is facial non-repetitively l-choosable. We use the so-called entropy compression method to show that π f l (G) ≤ c∆ for some absolute constant c and G a plane graph with maximum degree ∆. Moreover, we give some better (constant) upper bounds on π f l (G) for special classes of plane graphs.

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A sequence is called non-repetitive if none of its subsequences forms a repetition (a sequence r 1 r 2 • • • r 2n such that r i = r n+i for all 1 ≤ i ≤ n). Let G be a graph whose vertices are coloured. A colouring ϕ of the graph G is non-repetitive if the sequence of colours on every path in G is non-repetitive. The Thue chromatic number, denoted by π(G), is the minimum number of colours of a non-repetitive colouring of G.

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