On nonrepetitive colorings of cycles

Abstract

We say that a sequence a1 . a2t of integers is repetitive if ai = ai+t for every i ϵ {1,...,t}. A walk in a graph G is a sequence v1 . vr of vertices of G in which vivi+1 ϵ E(G) for every i ϵ {1,..., r - 1}. Given a k-coloring c: V(G) → {1,..., k} of V(G), we say that c is walk-nonrepetitive if for every t ϵ N, for every walk v1 . v2t in G the sequence c(V1) . c(v2t) is not repetitive unless vi = vi+t for every i ϵ {1,..., t}, and the walk-nonrepetitive chromatic number σ(G) of G is the minimum k for which G has a walk-nonrepetitive k-coloring. Let Cn denote the cycle with n vertices. In this paper we show that σ(Cn) = 4 whenever n ≥ 4 and n ∉ {5,7}, which answers a question posed by Barát and Wood in 2008.