On the facial Thue choice index of plane graphs

Let $G$ be a plane graph, and let $\phi$ be a colouring of its edges. The edge colouring $\phi$ of $G$ is called facial non-repetitive if for no sequence $r_1,r_2,\dots ,r_{2n}$, $n\ge 1$, of consecutive edge colours of any facial path we have $r_i=r_{n+i}$ for all $i=1,2,\dots ,n$. Assume that each edge $e$ of a plane graph $G$ is endowed with a list $L\left(e\right)$ 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 ${\pi }_{fl}\left(G\right)$. In this article we show that ${\pi }_{fl}^{\prime }\left(G\right)\le 291$ for arbitrary plane graphs $G$. Moreover, we give some better bounds for special classes of plane graphs.