Avoiding large squares in trees and planar graphs

TL;DR

This paper investigates the Thue number, a graph coloring parameter avoiding certain repetitive patterns, and extends it to generalized versions, providing bounds for trees and planar graphs.

Contribution

It introduces generalized Thue parameters _k(C) for graph classes and establishes new bounds for trees and planar graphs.

Findings

_5(tree)=2

_2(tree)=3

_k(planar)11 for all fixed k

Abstract

The Thue number $\pi(G)$ of a graph $G$ is the minimum number of colors needed to color $G$ without creating a square on a path of $G$. For a graph class $C$, $\pi(C)$ is the supremum of $\pi(G)$ over the graphs $G\in C$. The Thue number has been investigated for famous minor-closed classes: $\pi(tree)=4$, $7\le\pi(outerplanar)\le12$, and $11\le\pi(planar)\le768$. Following a suggestion of Grytczuk, we consider the generalized parameters $\pi_k(C)$ such that only squares of period at least $k$ must be avoided. Thus, $\pi(C)=\pi_1(C)$. We show that $\pi_5(tree)=2$, $\pi_2(tree)=3$, and $\pi_k(planar)\ge11$ for every fixed $k$.