PREPRINT

# Computing Square Colorings on Bounded-Treewidth and Planar Graphs

Akanksha Agrawal, Dániel Marx, Daniel Neuen, Jasper Slusallek

Submitted on 8 November 2022

## Abstract

A square coloring of a graph $G$ is a coloring of the square ${G}^{2}$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity of finding a square coloring with a given number of $q$ colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time ${n}^{{2}^{\mathrm{tw}+4}+O\left(1\right)}$ for graphs of treewidth at most $\mathrm{tw}$. The somewhat unusual exponent ${2}^{\mathrm{tw}}$ in the running time is essentially optimal: we show that for any $ϵ>0$, there is no algorithm with running time $f\left(\mathrm{tw}\right){n}^{\left(2-ϵ{\right)}^{\mathrm{tw}}}$ unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number $q\ge 4$ of colors. Our main algorithmic result is showing that the problem (when the number of colors $q$ is part of the input) can be solved in subexponential time ${2}^{O\left({n}^{2/3}\mathrm{log}n\right)}$ on planar graphs. The result follows from the combination of two algorithms. If the number $q$ of colors is small ($\le {n}^{1/3}$), then we can exploit a treewidth bound on the square of the graph to solve the problem in time ${2}^{O\left(\sqrt{qn}\mathrm{log}n\right)}$. If the number of colors is large ($\ge {n}^{1/3}$), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time ${2}^{O\left(n\mathrm{log}n/q\right)}$.

## Preprint

Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 2023

Subjects: Computer Science - Data Structures and Algorithms; Computer Science - Discrete Mathematics