Nonrepetitive graph colouring

Abstract

A vertex colouring of a graph G is nonrepetitive if G contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems.

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

“…Different authors successively improved the upper bounds on the nonrepetitive chromatic number and the nonrepetitive chromatic index and the best known bound for the nonrepetitive chromatic number is also in O(∆ 2 ) [2,4,10,14]. Non-repetitive colorings have since been studied in many other contexts (see for instance [20] for a recent survey on this topic).…”

“…For instance, every planar graph has nonrepetitive chromatic number at most 768, but can have an arbitrarily large nonrepetitive list chromatic number [3]. However, the best known bound on the nonrepetitive chromatic number in terms of the maximal degree also holds for the nonrepetititive list chromatic number [20]. It is unknown whether the optimal bounds in terms of the maximal degree also holds for the nonrepetititive list chromatic number are indeed identical or not.…”

“…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).…”

Nonrepetitively 3-colorable subdivisions of graphs with a logarithmic number of subdivisions per edge

“…Many generalizations and variations of this notion have been studied. In particular, the notion of nonrepetitive coloring of graphs was introduced by Alon et al [1] (see [16] for a recent survey on this topic). We say that a coloring (either of the vertices or of the edges) of a graph is nonrepetitive if the sequence of colors of any path is nonrepetitive.…”

“…Their result is in fact stronger than that since it holds in the list-coloring setting and that their bound is in fact O(log ∆) where ∆ is the maximal degree of the graph. Finally, Wood proved that every graph has a nonrepetitively 5colorable subdivision with O(log π(G)) division vertices per edge [16]. It is slightly stronger since it implies the same bound of O(log ∆), but it does not hold in the list-coloring setting and requires all the edges to be subdivided in the same amount of internal vertices.…”