Computing Square Colorings on Bounded-Treewidth and Planar Graphs

A square coloring of a graph G is a coloring of the square G² 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^{2tw +4+O(1)} for graphs of treewidth at most tw. The somewhat unusual exponent 2^tw in the running time is essentially optimal: we show that for any ε > 0, there is no algorithm with running time f(tw)n^(2-ε)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 ≥ 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(n2/3 log n)} on planar graphs. The result follows from the combination of two algorithms. If the number q of colors is small (≤ n^1/3), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2^{O(√qn log n)}.