Computing Square Colorings on Bounded-Treewidth and Planar Graphs

Abstract: A square coloring of a graph $G$ is a coloring of the square $G<sup>2$ 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 $$n^{{2^{{\operatorname{tw}}} + 4}+O(1)}$$ for graphs of treewidth at most $\operatorname{tw}$. The somewhat unusual exponent $2<sup>{\operatorname{tw}}$ in the running time is essentially optimal: we show that for any $\epsilon&gt;0$, there is no algorithm with running time $$f(\operatorname{tw})n<sup>{(2-\epsilon)<sup>{\operatorname{tw}}}$$ 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 $2^{{O(n^{2/3}\log} n)}$ on planar graphs. The result follows from the combination of two algorithms. If the number $q$ of colors is small ($\le n<sup>{1/3}$), then we can exploit a treewidth bound on the square of the graph to solve the problem in time $2^{{O(\sqrt{qn}\log} n)}$. If the number of colors is large ($\ge n<sup>{1/3}$), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time $2^{O(n\log n/q)}$.