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On Distance Coloring

A Review Based on Work with Dexter Kozen
  • Alexa Sharp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7230)

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].

Keywords

Tree Decomposition Simple Cycle Articulation Point Cycle Structure Tutte Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexa Sharp
    • 1
  1. 1.Oberlin CollegeOberlinUSA

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