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Distributed Vertex Coloring

2004; Finocchi, Panconesi, Silvestri

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Encyclopedia of Algorithms

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Keywords and Synonyms

Vertex coloring; Distributed computation

Problem Definition

The vertex coloring problem takes as input an undirected graph \( { G:=(V,E) } \) and computes a vertex coloring, i. e. a function, \( { c: V \rightarrow [k] } \) for some positive integer k such that adjacent vertices are assigned different colors (that is, \( { c(u) \not = c(v) } \) for all \( { (u,v) \in E } \)). In the \( { (\Delta + 1) } \) vertex coloring problem, k is set equal to \( { \Delta +1 } \) where Δ is the maximum degree of the input graph G. In general, \( { (\Delta +1) } \) colors could be necessary as the example of a clique shows. However, if the graph satisfies certain properties, it may be possible to find colorings with far fewer colors. Finding the minimum number of colors possible is a computationally hard problem: the corresponding decision problems are NP-complete [5]. In Brooks–Vizing colorings, the goal is to try to find colorings that are near optimal.

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Recommended Reading

  1. Alon, N., Spencer, J.: The Probabilistic Method. Wiley (2000)

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  2. Culberson, J.C.: http://web.cs.ualberta.ca/~joe/Coloring/index.html

  3. Ftp site of DIMACS implementation challenges, ftp://dimacs.rutgers.edu/pub/challenge/

  4. Finocchi, I., Panconesi, A., Silvestri, R.: An experimental Analysis of Simple Distributed Vertex Coloring Algorithms. Algorithmica 41, 1–23 (2004)

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  5. Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman (1979)

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  6. Grable, D.A., Panconesi, A.: Fast distributed algorithms for Brooks–Vizing colorings. J. Algorithms 37, 85–120 (2000)

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  7. Johansson, Ö.: Simple distributed \( { (\Delta + 1) } \)-coloring of graphs. Inf. Process. Lett. 70, 229–232 (1999)

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  8. Kim, J.-H.: On Brook's Theorem for sparse graphs. Combin. Probab. Comput. 4, 97–132 (1995)

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  9. Luby, M.: Removing randomness in parallel without processor penalty. J. Comput. Syst. Sci. 47(2), 250–286 (1993)

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  10. Molly, M., Reed, B.: Graph Coloring and the Probabilistic method. Springer (2002)

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  11. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. In: SIAM Monographs on Discrete Mathematics and Applications 5 (2000)

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  12. Trick, M.: Michael Trick's coloring page: http://mat.gsia.cmu.edu/COLOR/color.html

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Author information

Authors and Affiliations

  1. Department of Computer Science, Chalmers University of Technology and Gothenburg University, Gothenburg, Sweden

    Devdatt Dubhashi

Authors
  1. Devdatt Dubhashi
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Dubhashi, D. (2008). Distributed Vertex Coloring. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_118

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