Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Exact Graph Coloring Using Inclusion-Exclusion

  • Andreas BjörklundEmail author
  • Thore Husfeldt
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_134

Years and Authors of Summarized Original Work

  • 2006; Björklund, Husfeldt

Problem Definition

A k-coloring of a graph G = (V, E) assigns one of k colors to each vertex such that neighboring vertices have different colors. This is sometimes called vertex coloring.

The smallest integer k for which the graph G admits a k-coloring is denoted χ(G) and called the chromatic number. The number of k-colorings of G is denoted P(G; k) and called the chromatic polynomial.

Key Results

The central observation is that χ(G) and P(G; k) can be expressed by an inclusion-exclusion formula whose terms are determined by the number of independent sets of induced subgraphs of G. For \(X \subseteq V\)


Vertex coloring 
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Recommended Reading

  1. 1.
    Björklund A, Husfeldt T (2008) Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2):226–249MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Björklund A, Husfeldt T, Kaski P, Koivisto M (2007) Fourier meets Möbius: fast subset convolution. In: Proceedings of the 39th annual ACM symposium on theory of computing (STOC), San Diego, 11–13 June 2007. Association for Computing Machinery, New York, pp 67–74Google Scholar
  3. 3.
    Björklund A, Husfeldt T, Koivisto M (2009) Set partitioning via inclusion-exclusion. SIAM J Comput 39(2):546–563MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Björklund A, Husfeldt T, Kaski P, Koivisto M (2011) Covering and packing in linear space. Inf Process Lett 111(21–22):1033–1036MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceLund UniversityLundSweden