Pathwidth and nonrepetitive list coloring

TL;DR

This paper investigates nonrepetitive list coloring of graphs, proving that trees with bounded pathwidth are nonrepetitively choosable with a function depending on pathwidth, but this property does not extend to all graphs.

Contribution

It establishes a positive result for trees with bounded pathwidth and shows the limitation of this property for more general graphs.

Findings

Every tree of pathwidth k is nonrepetitively f(k)-choosable.

There exists a family of pathwidth-2 graphs not nonrepetitively $oldsymbol{ ext{l}}$-choosable for any fixed $ ext{l}$.

Abstract

A vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every $\ell \geq 1$ there is a tree that is not nonrepetitively $\ell$-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function $f$ such that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed $\ell$.