Skip to main content
SpringerLink
Log in

The Complexity of Graph Ramsey Games

  • Conference paper
  • First Online:
Computers and Games (CG 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2063))

Included in the following conference series:

Abstract

We consider combinatorial avoidance and achievement games based on graph Ramsey theory: The players take turns in coloring edges of a graph G, each player being assigned a distinct color and choosing one so far uncolored edge per move. In avoidance games, completing a monochromatic subgraph isomorphic to another graph A leads to immediate defeat or is forbidden and the first player that cannot move loses. In the avoidance+ variant, both players are free to choose more than one edge per move. In achievement games, the first player that completes a monochromatic subgraph isomorphic to A wins. We prove that general graph Ramsey avoidance, avoidance+, and achievement endgames and several variants thereof are PSPACE-complete.

Acknowledgments

This research was partially supported by Austrian Science Fund Project N Z29-INF. I am grateful to the anonymous reviewers for their valuable comments.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. J. Beck and L. Csirmaz. Variations on a game. Journal of Combinatorial Theory, Series A, 33:297–315, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  2. József Beck. Van derWaerden and Ramsey type games. Combinatorica, 1(2):103–116, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  3. József Beck. Biased Ramsey type games. Studia Scientiarum Mathematicarum Hungarica, 18:287–292, 1983.

    MATH  MathSciNet  Google Scholar 

  4. József Beck. Graph games. In Proceedings of the International Colloquium on Extremal Graph Theory, 1997.

    Google Scholar 

  5. Peter J. Cameron. Combinatorics: topics, techniques, algorithms. Cambridge University Press, 1994.

    Google Scholar 

  6. V. Chvátal and Frank Harary. Generalized Ramsey theory for graphs. Bull. Amer. Math. Soc., 78:423–426, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  7. Michael L. Cook and Leslie E. Shader. A strategy for the Ramsey game “Tritip”. In Proc. 10th southeast. Conf. Combinatorics, Graph Theory and Computing, Boca Raton, volume 1 of Congr. Numerantium 23, pages 315–324, 1979.

    Google Scholar 

  8. Michael Cornelius and Alan Parr. What’s your game?: A resource book for mathematical activities. Cambridge University Press, 1991.

    Google Scholar 

  9. Carl Darby and Richard Laver. Countable length Ramsey games. In Carlos Augusto Di Prisco et al., editor, Set theory: techniques and applications. Proceedings of the conferences, Curacao, Netherlands Antilles, June 26-30, 1995 and Barcelona, Spain, June 10-14, 1996, pages 41–46. Dordrecht: Kluwer Academic Publishers, 1998.

    Google Scholar 

  10. A. P. DeLoach. Some investigations into the game of SIM. J. Recreational Mathematics, 4(1):36–41, January 1971.

    Google Scholar 

  11. W. Deuber.A generalization of Ramsey’s theorem. In A. Hajnal, R. Radó, and V.T. Sós, editors, Infinite and finite sets, volume 10 of Colloq. Math. Soc. János Bolyai, pages 323–332. North-Holland, 1975.

    Google Scholar 

  12. Douglas Engel. DIM: three-dimensional Sim. J. Recreational Mathematics, 5:274–275, 1972.

    MathSciNet  Google Scholar 

  13. P. Erdos and J. L. Selfridge. On a combinatorial game. Journal of Combinatorial Theory, Series A, 14:298–301, 1973.

    Article  MathSciNet  Google Scholar 

  14. Paul Erdos, András Hajnal, and Latjos Pósa. Strong embeddings of graphs into colored graphs. In A. Hajnal, R. Radó, and V.T. Sós, editors, Infinite and finite sets, volume 10 of Colloq. Math. Soc. János Bolyai, pages 585–595. North-Holland, 1975.

    Google Scholar 

  15. Geoffrey Exoo. A new way to play Ramsey games. J. Recreational Mathematics, 13(2):111–113, 1980-81.

    MATH  MathSciNet  Google Scholar 

  16. A. S. Fraenkel and D. Lichtenstein. Computing a perfect strategy for n×n Chess requires time exponential in n. Journal of Combinatorial Theory, Series A, 31:199–214, 1981. Preliminary version in Proc. 8th Internat. Colloq. Automata, Languages and Programming (S. Even and O. Kariv, eds.), Acre, Israel, 1981, Lecture Notes in Computer Science 115, 278-293, Springer Verlag, Berlin.

    Google Scholar 

  17. Aviezri S. Fraenkel. Complexity of games. In Richard K. Guy, editor, Combinatorial Games, volume 43 of Proceedings of Symposia in Applied Mathematics, pages 111–153. American Mathematical Society, 1991.

    Google Scholar 

  18. Aviezri S. Fraenkel. Dynamic surveys in combinatorics: Combinatorial games: Selected bibliography with a succinct gourmet introduction. Electronic Journal of Combinatorics, 1994 (revised January 1, (2000).

    Google Scholar 

  19. W.W. Funkenbusch. SIM as a game of chance. J. Recreational Mathematics, 4(4):297–298, October 1971.

    Google Scholar 

  20. Martin Gardner. Mathematical games. Scientific American, 228(1):108–115, January 1973.

    Article  Google Scholar 

  21. Martin Gardner. Mathematical games. Scientific American, 228(5):102–107, May 1973.

    Google Scholar 

  22. Martin Gardner. Knotted doughnuts and other mathematical entertainments, chapter 9, pages 109–122. W. H. Freeman and Company, 1986.

    Google Scholar 

  23. Michael R. Garey and David S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. Freeman and Co., 1979.

    Google Scholar 

  24. Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer. Ramsey theory. Wiley, 2nd edition, 1990.

    Google Scholar 

  25. A. Hajnal and Zs. Nagy. Ramsey games. Transactions of the American Mathematical Society, 284(2):815–827, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  26. Frank Harary. Achievement and avoidance games for graphs. In Béla Bollobás, editor, Proceedings of the Conference on Graph Theory, volume 13 of Ann. Discrete Math., pages 111–119, Cambridge, 1982. North-Holland, mathematics studies 62.

    Google Scholar 

  27. Frank Harary, Wolfgang Slany, and Oleg Verbitsky. A symmetric strategy in graph avoidance games. Manuscript presented at the Combinatorial Game Theory Research Workshop held at the Mathematical Sciences Research Institute in Berkeley, California, July 2000.

    Google Scholar 

  28. S. Iwata and T. Kasai. The Othello game on an n × n board is PSPACE-complete. Theoret. Comput. Sci. (Math Games), 123:329–340, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  29. Martin Knor. On Ramsey-type games for graphs. Australasian J. Comb., 14:199–206, 1996.

    MATH  MathSciNet  Google Scholar 

  30. Péter Komjáth. Asimple strategy for the Ramsey-game. Studia Scientiarum Mathematicarum Hungarica, 19:231–232, 1984.

    MATH  MathSciNet  Google Scholar 

  31. D. Lichtenstein and M. Sipser. Go is polynomial-space hard. J. Assoc. Comput. Mach., 27:393–401, 1980. Earlier draft appeared in Proc. 19th Ann. Symp. Foundations of Computer Science (Ann Arbor, MI, Oct. 1978), IEEE Computer Soc., Long Beach, CA, 1978, 48-54.

    Google Scholar 

  32. Ernest Mead, Alexander Rosa, and Charlotte Huang. The game of SIM: A winning strategy for the second player. Mathematics Magazine, 47(5):243–247, November 1974.

    MATH  MathSciNet  Google Scholar 

  33. Thomas E. Moore. SIM on a microcomputer. J. Recreational Mathematics, 19(1):25–29, 1987.

    Article  Google Scholar 

  34. John H. Nairn and A. B. Sperry. SIM on a desktop calculator. J. Recreational Mathematics, 6(4):243–251, Fall 1973.

    Google Scholar 

  35. Jaroslav Nešetřil. Ramsey theory. In R.L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics, chapter 25, pages 1331–1403. Elsevier, 1995.

    Google Scholar 

  36. G. L. O’Brian. The graph of positions in the game of SIM. J. Recreational Mathematics, 11(1):3–9, 1978-79.

    Google Scholar 

  37. Christos H. Papadimitriou. Games against Nature. Journal of Computer and System Sciences, 31:288–301, 1985.

    MATH  Google Scholar 

  38. Christos H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.

    Google Scholar 

  39. Aleksandar Pekeč. A winning strategy for the Ramsey graph game. Combinatorics, Probability & Computing, 5(3):267–276, September 1996.

    MATH  Google Scholar 

  40. Frank P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30(2):264–286, 1930. Reprinted in I. Gessel and G.-C. Rota, editors. Classic Papers in Combinatorics. Pages 2-24. Birkhäuser Boston 1987.

    Article  Google Scholar 

  41. Duncan C. Richer. Maker-breaker games. Poster-presentation at Erdos’99 July 4-11, 1999, in Budapest, Hungary, 1999.

    Google Scholar 

  42. Vojtech Rödl. A generalization of Ramsey theorem and dimension of graphs. Msc thesis, Charles University, Prague, 1973.

    Google Scholar 

  43. E. M. Rounds and S. S. Yau. A winning strategy for SIM. J. Recreational Mathematics, 7(3):193–202, Summer 1974.

    Google Scholar 

  44. Thomas J. Schaefer. On the complexity of some two-person perfect-information games. Journal of Computer and System Sciences, 16(2):185–225, April 1978.

    Article  MATH  MathSciNet  Google Scholar 

  45. Benjamin L. Schwartz, editor. Mathematical solitaires and games, pages 37–81. Baywood Publishing Company, Farmingdale, NY, 1979.

    Google Scholar 

  46. Benjamin L. Schwartz. SIM with non-perfect players. J. Recreational Mathematics, 14(4):261–265, 1981-82.

    Google Scholar 

  47. Leslie E. Shader. Another strategy for SIM. Mathematics Magazine, 51(1):60–64, January 1978.

    Article  MATH  MathSciNet  Google Scholar 

  48. Leslie E. Shader and Michael L. Cook. A winning strategy for the second player in the game Tri-tip. In Proc. Tenth S.E. Conference on Computing, Combinatorics and Graph Theory, Utilitas,Winnipeg, 1980.

    Google Scholar 

  49. Gustavus J. Simmons. The game of SIM. J. Recreational Mathematics, 2(2):66, April 1969.

    Google Scholar 

  50. Wolfgang Slany. HEXI: A happy perfect hexagone player. Technical Report 31, Institut für Statistik und Informatik der UniversitôtWien, January 1988.

    Google Scholar 

  51. Wolfgang Slany. Graph Ramsey games. Technical Report DBAI-TR-99-34, Institut für Informationssysteme der Technischen UniversitätWien, 1999.

    Google Scholar 

  52. Bent Thomsen and Lone Leth Thomsen. Towards global computations guided by concurrency theory. Bulletin of the European Association for Theoretical Computer Science, (66):92–98, October 1998.

    MATH  MathSciNet  Google Scholar 

  53. Bidan Zhu. Ramsey theory illustrated through a Java based game that plays heuristically and can learn in a client-server style or optimally if possible. Master’s thesis, Technische UniversitätWien, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informationssysteme, Technische Universität Wien, Favoritenstr. 9-11, A-1040, Wien, Austria

    Wolfgang Slany

Authors
  1. Wolfgang Slany
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Alberta, Edmonton, Alberta, Canada

    Tony Marsland

  2. Future University - Hakodate, Hakodate, Hokkaido, Japan

    Ian Frank

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Slany, W. (2001). The Complexity of Graph Ramsey Games. In: Marsland, T., Frank, I. (eds) Computers and Games. CG 2000. Lecture Notes in Computer Science, vol 2063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45579-5_12

Download citation

  • DOI: https://doi.org/10.1007/3-540-45579-5_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43080-3

  • Online ISBN: 978-3-540-45579-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation