$$(2+\epsilon )$$ ( 2 + ϵ ) -Nonrepetitive List Colouring of Paths

Abstract

A sequence a 1 a 2 . . . a p is an r -repetition (for a real number r > 1) if p = rq for some positive integer q, and a j = a j+q for j = 1, 2, . . . , p − q. In other words, the sequence can be divided into r blocks where all the blocks are the same, say, all the blocks equal to a 1 a 2 . . . a q for some q ≥ 1, except that when r is not an integer, the last block is the prefix of a 1 ...a q of length (r − r )q . A colouring of the vertices of a graph G is r -nonrepetitive if there is no path in G for which the colour sequence of its vertices forms an r -repetition. The r -nonrepetitive chromatic number π r (G) of G is the minimum number of colours needed in an r -nonrepetitive colouring of G. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The r -nonrepetitive choice number π rch (G) of G is the least integer k such that for every k-list assignment L, there is an r -nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. A classical result of Thue asserts that π 2 (P n ) ≤ 3 for all n. It is known that π 2ch (P n ) ≤ 4 for all n. However, it remains an open problem whether π 2ch (P n ) ≤ 3 for all n. This paper proves that for any > 0, π (2+ )ch (P n ) ≤ 3 for all n.

Nonrepetitive Graph Colouring

“…Note that a simple adaptation to the proof of Theorem 3.6 shows that every path is list 3-colourable such that every subpath with at least four vertices is nonrepetitively coloured; that is, the only repetitively coloured subpaths have two vertices. The results of Zhao and Zhu [152] are also relevant here.…”

Nonrepetitive graph colouring

“…Note that a simple adaptation to the proof of Theorem 3.6 shows that every path is list 3colourable such that every subpath with at least four vertices is nonrepetitively coloured; that is, the only repetitively coloured subpaths have two vertices. The results of Zhao and Zhu [149] are also relevant here.…”

Avoiding squares over words with lists of size three amongst four symbols

“…It was first conjectured by Grytczuk [6] that the nonrepetitive list chromatic number of any path is in fact at most 3. This conjecture has been mentioned many times, but not much progress has been made in the direction of proving or disproving it (see for instance [5,6,7,9,13,14,16,20,21] for some of the mentions of this problem).…”