On the facial Thue choice index of plane graphs

Let G be a plane graph, and let φ be a colouring of its edges. The edge colouring φ of G is called facial non-repetitive if for no sequence r 1 , r 2 , … , r 2 n , n ≥ 1 , of consecutive edge colours of any facial path we have r i = r n + i for all i = 1 , 2 , … , n . Assume that each edge e of a plane graph G is endowed with a list L ( e ) of colours, one of which has to be chosen to colour e . The smallest integer k such that for every list assignment with minimum list length at least k there exists a facial non-repetitive edge colouring of G with colours from the associated lists is the facial Thue choice index of G , and it is denoted by π f l ( G ) . In this article we show that π f l ′ ( G ) ≤ 291 for arbitrary plane graphs G . Moreover, we give some better bounds for special classes of plane graphs.