On Distance Coloring

Abstract

An undirected graph G = (V,E) is (d,k)-colorable if there is a vertex coloring using at most k colors such that no two vertices within distance d have the same color. It is well known that (1,2)-colorability is decidable in linear time, and that (1,k)-colorability is NP-complete for k ≥ 3. This paper presents the complexity of (d,k)-coloring for general d and k, and enumerates some interesting properties of (d,k)-colorable graphs. The main result is the dichotomy between polynomial and NP-hard instances: for fixed d ≥ 2, the distance coloring problem is polynomial time for $k \leq \lfloor \frac{3d}{2} \rfloor$ and NP-hard for $k > \lfloor \frac{3d}{2} \rfloor$. We present a reduction in the latter case, as well as an algorithm in the former. The algorithm entails several innovations that may be of independent interest: a generalization of tree decompositions to overlay graphs other than trees; a general construction that obtains such decompositions from certain classes of edge partitions; and the use of homology to analyze the cycle structure of colorable graphs. This paper is both a combining and reworking of the papers of Sharp and Kozen [14, 10].