On the facial Thue choice index of plane graphs

Let GG be a plane graph, and let φφ be a colouring of its edges. The edge colouring φφ of GG is called facial non-repetitive if for no sequence r1,r2,…,r2nr1,r2,…,r2n, n≥1n≥1, of consecutive edge colours of any facial path we have ri=rn+iri=rn+i for all i=1,2,…,ni=1,2,…,n. Assume that each edge ee of a plane graph GG is endowed with a list L(e)L(e) of colours, one of which has to be chosen to colour ee. The smallest integer kk such that for every list assignment with minimum list length at least kk there exists a facial non-repetitive edge colouring of GG with colours from the associated lists is the facial Thue choice index of GG, and it is denoted by πfl(G)πfl(G). In this article we show that πfl′(G)≤291 for arbitrary plane graphs GG. Moreover, we give some better bounds for special classes of plane graphs.