Facial anagram-free edge-coloring of plane graphs

A sequence r1,r2,…,r2n is called an anagram if rn+1,rn+2,…,r2n is a permutation of r1,r2,…,rn. A sequence S is called anagram-free if no block (i.e. subsequence of consecutive terms of S) is an anagram. A coloring of the edges of a given plane graph G is called facial anagram-free if the sequence of colors on any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) in G is anagram-free. In this paper we show that every connected plane graph G admits a facial anagram-free edge-coloring with at most 11 colors. Moreover, if G is a 3-connected plane graph, then 9 colors suffice for such a coloring.