Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Combinatorial Algorithms

IWOCA 2017: Combinatorial Algorithms pp 244–251Cite as

On Solving the Queen Graph Coloring Problem

  • Conference paper
  • First Online:

  • 632 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10765))

Abstract

The chromatic numbers of queen\(\,-\,\) \({{n}^2}\) graphs are difficult to determine when n > 9 and n is a multiple of 2 or 3. In previous works [6, 7], we have proven that this number (denoted \(\chi _{n}\,\)) is equal to n for \(n \in \{12,14,15,16,18,20,21,22,24,28,32\}\) and that \(\chi _{10}\,\)\(=11\). This article describes how, by extending slightly further the previous work, the chromatic number of queen\(\,-\,\) \({{26}^2}\) and queen\(\,-\,\) \({{30}^2}\) can be obtained. A more general result, proving that \(\chi _{2n}\,\)\(=2n\) and \(\chi _{3n}\,\)\(=3n\) for infinitely many values of n, is then presented.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    http://ww.cs.concordia.ca/~chvatal/queengraphs.html.

  2. 2.

    pentium4 2.4 Ghz cpu as mentioned above.

References

  1. Abramson, B., Yung, M.M.: Construction through decomposition: a divide-and-conquer algorithm for the N-queens problem. In: Proceedings of 1986 ACM Fall Joint Computer Conference. ACM 1986, pp. 620–628, no. 9. IEEE Computer Society Press, Los Alamitos (1986). http://dl.acm.org/citation.cfm?id=324493.324620.

  2. Caramia, M., Dell’Olmo, P.: Embedding a novel objective function in a two-phased local search for robust vertex coloring. Eur. J. Oper. Res. 189(3), 1358–1380 (2008). https://doi.org/10.1016/j.ejor.2007.01.063. 9

    Article  MathSciNet  MATH  Google Scholar 

  3. Galinier, P., Hertz, A., Zufferey, N.: An adaptive memory algorithm for the k-coloring problem. Discrete Appl. Math. 156(2), 267–279 (2008). https://doi.org/10.1016/j.dam.2006.07.017. Computational Methods for Graph Coloring and it’s Generalizations. http://www.sciencedirect.com/science/article/pii/S0166218X07001114

    Article  MathSciNet  MATH  Google Scholar 

  4. Gualandi, S., Malucelli, F.: Exact solution of graph coloring problems via constraint programming and column generation. INFORMS J. Comput. 24(1), 81–100 (2012). https://doi.org/10.1287/ijoc.1100.0436

    Article  MathSciNet  MATH  Google Scholar 

  5. Hansen, P., Labbé, M., Schindl, D.: Set covering and packing formulations of graph coloring: algorithms and first polyhedral results. Discrete Optim. 6(2), 135–147 (2009). https://doi.org/10.1016/j.disopt.2008.10.004. http://www.sciencedirect.com/science/article/pii/S1572528608000716

    Article  MathSciNet  MATH  Google Scholar 

  6. Vasquez, M.: New results on the queens_n2 graph coloring problem. J. Heuristics 10(4), 407–413 (2004). https://doi.org/10.1023/B:HEUR.0000034713.28244.e1

    Article  Google Scholar 

  7. Vasquez, M., Habet, D.: Complete and incomplete algorithms for the queen graph coloring problem. In: Proceedings of the 16th European Conference on Artificial Intelligence, ECAI 2004, including Prestigious Applicants of Intelligent Systems, PAIS 2004, Valencia, Spain, 22–27 August, 2004, pp. 226–230 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole des mines d’Alès, Alès, France

    Michel Vasquez & Yannick Vimont

Authors
  1. Michel Vasquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yannick Vimont
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Vasquez .

Editor information

Editors and Affiliations

  1. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Ljiljana Brankovic

  2. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Joe Ryan

  3. McMaster University, Hamilton, Ontario, Canada

    William F. Smyth

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Vasquez, M., Vimont, Y. (2018). On Solving the Queen Graph Coloring Problem. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78825-8_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78824-1

  • Online ISBN: 978-3-319-78825-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation