On distance coloring

• Authors: Alexa Sharp
• Venue: Logic and Program Semantics
• Year: 2012

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].