On the facial Thue choice index of plane graphs

Abstract

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 r1,r2,…,r2n, n≥1, of consecutive edge colours of any facial path we have ri=rn+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 πfl(G). In this article we show that πfl′(G)≤291 for arbitrary plane graphs G. Moreover, we give some better bounds for special classes of plane graphs.