Abstract. A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge colouring of a path is nonrepetitive if the sequence of colours of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colourings for arbitrarily long paths using only three colours. A recent generalization of this concept implies that we may obtain such colourings even if we are forced to choose edge colours from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Soták anď Skrabul'áková proved that for each plane graph, 8 colours are sufficient to provide an edge colouring so that every facial path is nonrepetitively coloured. In this paper we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovász Local Lemma). Our approach is based on the Moser-Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik and Micek, and by Dujmović, Joret, Kozik and Wood.
For the list-colouring version of this problem, Przybyło REF showed that lists with at least 12 colours are sufficient to colour the edges of any plane graph so that every facial trail is coloured nonrepetitively.
On the facial Thue choice index via entropy compression
J. Graph Theory 77(3) (2014) 180-189
Mathematics Computer Science