Nonrepetitive choice number of trees

TL;DR

This paper investigates the nonrepetitive choice number of trees, establishing an almost linear upper bound in maximum degree and contrasting it with known lower bounds, while also exploring color repetitions.

Contribution

It introduces an almost linear bound on the Thue choice number for trees, improving previous quadratic bounds for graphs with bounded degree.

Findings

Almost linear bound $oxed{ ext{c}\Delta^{1+ ext{ε}}}$ for trees' nonrepetitive choice number

Existence of trees with logarithmic lower bounds on the choice number

Repetition allowance with bounded lists enables nonrepetitive coloring of any tree

Abstract

A nonrepetitive coloring of a path is a coloring of its vertices such that the sequence of colors along the path does not contain two identical, consecutive blocks. The remarkable construction of Thue asserts that 3 colors are enough to color nonrepetitively paths of any length. A nonrepetitive coloring of a graph is a coloring of its vertices such that all simple paths are nonrepetitively colored. Assume that each vertex $v$ of a graph $G$ has assigned a set (list) of colors $L_v$. A coloring is chosen from $\{L_v\}_{v\in V(G)}$ if the color of each $v$ belongs to $L_v$. The Thue choice number of $G$, denoted by $\pi_l(G)$, is the minimum $k$ such that for any list assignment $\set{L_v}$ of $G$ with each $|L_v|\geq k$ there is a nonrepetitive coloring of $G$ chosen from $\{L_v\}$. Alon et al. (2002) proved that $\pi_l(G)=O(\Delta^2)$ for every graph $G$ with maximum degree at most $\Delta$. We propose an almost linear bound in $\Delta$ for trees, namely for any $\epsi>0$ there is a constant $c$ such that $\pi_l(T)\leq c\Delta^{1+\epsi}$ for every tree $T$ with maximum degree $\Delta$. The only lower bound for trees is given by a recent result of Fiorenzi et al. (2011) that for any $\Delta$ there is a tree $T$ such that $\pi_l(T)=\Omega(\frac{\log\Delta}{\log\log\Delta})$. We also show that if one allows repetitions in a coloring but still forbid 3 identical consecutive blocks of colors on any simple path, then a constant size of the lists allows to color any tree.