A sequence r 1 , r 2 , . . . , r 2n such that r i = r n+i for all 1 ≤ i ≤ n is called a repetition. A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non-repetitive if the sequence of colors of its edges is non-repetitive. If G is a plane graph, a facial non-repetitive edge-coloring of G is an edge-coloring such that any facial

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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.

A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced