The Thue choice number versus the Thue chromatic number of graphs

TL;DR

This paper compares the Thue chromatic number and the Thue choice number of graphs, highlighting their differences across various graph families and discussing open problems in the area.

Contribution

It provides an overview of known results comparing these two parameters and discusses the potential for large differences in various graph classes.

Findings

Thue chromatic number and Thue choice number can differ arbitrarily in some graph classes.

Comparison of these parameters across multiple graph families is summarized.

Open problems related to these parameters are identified and discussed.

Abstract

We say that a vertex colouring $\varphi$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours $\varphi(v_{1})\ldots\varphi(v_{2n})$ satisfy $\varphi(v_{i})=\varphi(v_{i+n})$ for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive vertex colouring of $G$ is the Thue chromatic number $\pi (G)$. For the case of vertex list colourings the Thue choice number $\pi_{l}(G)$ of $G$ denotes the smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ with minimum list length at least $k$, there is a nonrepetitive vertex colouring of $G$ from the assigned lists. Recently it was proved that the Thue chromatic number and the Thue choice number of the same graph may have an arbitrary large difference in some classes of graphs. Here we give an overview of the known results where we compare these two parameters for several families of graphs and we also give a list of open problems on this topic.