On the Complexity of Some Map-Coloring Multi-player Games

  • Alessandro CincottiEmail author
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 52)


Col and Snort are two-player map-coloring games invented respectively by Colin Vout and Simon Norton where to establish who has a winning strategy on a general graph is a \(\mathcal{P}SPACE\)-complete problem. However, winning strategies can be found on specific graph instances, e.g., strings or trees. In multi-player games, because of the possibility to form alliances, cooperation between players is a key-factor to determine the winning coalition and, as a result, the complexity of three-player Col played on trees is \(\mathcal{N}\mathcal{P}\)-complete and the complexity of n-player Snort played on bipartite graphs is \(\mathcal{P}\mathcal{S}\mathcal{P}\mathcal{A}\mathcal{C}\mathcal{E}\)-complete.


Complexity Map-coloring game Multi-player game 



The author wishes to thank Mark G. Elwell for a careful reading of the manuscript.


  1. 1.
    Berlekamp, E.R., Conway, J.H., & Guy, R.K. (2001). Winning way for your mathematical plays. Natick, MA: AK Peters.Google Scholar
  2. 2.
    Cincotti, A. (2005). Three-player partizan games. Theoretical Computer Science, 332, 367–389.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cincotti, A. (2007). Counting the number of three-player partizan cold games. In H.J. van den Herik, P. Ciancarini, & H.H.L.M. Donkers (Eds.), Computers and games, volume 4630 of LNCS (pp. 181–189). Springer, Germany.Google Scholar
  4. 4.
    Cincotti, A. (2008a). The game of cutblock. INTEGERS: Electronic Journal of Combinatorial Number Theory, 8(G06), 1–12.MathSciNetGoogle Scholar
  5. 5.
    Cincotti, A. (2008b). Three-player hackenbush played on strings is \(\mathcal{N}\mathcal{P}\)-complete. In S.I. Ao, O. Castillo, C. Douglas, D.D. Feng, & J. Lee (Eds.), International multiconference of engineers and computer scientists 2008 (pp. 226–230). Newswood Limited, Hong Kong.Google Scholar
  6. 6.
    Cincotti, A. (2009a). Further results on the game of col. In Proceedings of the 8th Annual Hawaii international conference on statistics, mathematics and related fields (pp. 315–318).Google Scholar
  7. 7.
    Cincotti, A. (2009b). Three-player col played on trees is \(\mathcal{N}\mathcal{P}\)-complete. In S.I. Ao, O. Castillo, C. Douglas, D.D. Feng, & J. Lee, (Eds.), International multiconference of engineers and computer scientists 2009 (pp. 445–447). Newswood Limited, Hong Kong.Google Scholar
  8. 8.
    Cincotti, A., & Bossart, T. (2008). The game of col on complete k-ary trees. In C. Ardil (Ed.), Proceedings of world academy of science, engineering and technology, volume 30 (pp. 699–701).Google Scholar
  9. 9.
    Conway, J.H. (2001). On numbers and games. Natick, MA: AK Peters.Google Scholar
  10. 10.
    Garey, M.R., & Johnson, D.S. (1979). Computers and intractability. New York: Freeman.zbMATHGoogle Scholar
  11. 11.
    Papadimitriou, C.H. (1994). Computational complexity. Reading, MA: Addison-Wesley.zbMATHGoogle Scholar
  12. 12.
    Propp, J.G. (2000). Three-player impartial games. Theoretical Computer Science, 233, 263–278.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Schaefer, T.J. (1978). On the complexity of some two-person perfect information games.Journal of Computer Systems and Science, 16(2), 185–225.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan

Personalised recommendations