Facial non-repetitive vertex colouring of some families of 2-connected plane graphs

A sequence a1a2 . . . a2n such that ai = an+i for all 1 ≤ i ≤ n is called a repetition. A sequence S is called non-repetitive if no subsequence of consecutive terms of S is a repetition. Let G be a graph whose vertices are coloured. A path in G is a called nonrepetitive if the sequence of colours of its vertices is non-repetitive. If G is a plane graph, a facial non-repetitive vertex colouring of G is a vertex colouring such that any facial path (i.e. a path of consecutive vertices on the boundary walk of a face) is non-repetitive. We denote πf (G) the minimum number of colours of a facial non-repetitive vertex colouring of G. In this article, we show that πf (G) ≤ 16 for any plane hamiltonian graph G and πf (G) ≤ 112 for any 2-connected cubic plane graph G. These bounds are improved for some subfamilies of 2-connected plane graphs. All proofs are constructive.