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arXiv · 2106.01521

Avoiding large squares in trees and planar graphs

Daniel Gonçalves·Pascal Ochem·Matthieu Rosenfeld
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Abstract

The Thue number $π(G)$ 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 $C$, $π(C)$ is the supremum of $π(G)$ over the graphs $G\in C$. The Thue number has been investigated for famous minor-closed classes: $π(tree)=4$, $7\leπ(outerplanar)\le12$, and $11\leπ(planar)\le768$. Following a suggestion of Grytczuk, we consider the generalized parameters $π_k(C)$ such that only squares of period at least $k$ must be avoided. Thus, $π(C)=π_1(C)$. We show that $π_5(tree)=2$, $π_2(tree)=3$, and $π_k(planar)\ge11$ for every fixed $k$.

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Daniel Gonçalves, Pascal Ochem, Matthieu Rosenfeld. 2021-06-03. Avoiding large squares in trees and planar graphs. https://arxiv.org/abs/2106.01521

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