Chomp on generalized Kneser graphs and others

  • Ignacio García-Marco
  • Kolja Knauer
  • Luis Pedro MontejanoEmail author
Original Paper


In Chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can an explicit strategy be devised? We answer these questions (and determine the Nim-value) for the class of generalized Kneser graphs and for several families of Johnson graphs. We also generalize some of these results to the clique complexes of these graphs. Furthermore, we determine which player has a winning strategy for some classes of threshold graphs.


Combinatorial games Impartial games Chomp Generalized Kneser graphs Johnson graphs Threshold graphs Clique complex 

Mathematics Subject Classification

05C57 91A05 91A46 91A43 91A05 



Both I.G. and K.K. thank L.P.M. and Mireia Ferrer for their hospitality and the good time spent during the stay in Guanajuato. We thank Cormac O’Sullivan for pointing out an error in a theorem about almost bipartite graphs stated in an earlier version of the paper.


  1. Berlekamp ER, Conway JH, Guy RK (2001) Winning ways for your mathematical plays, 2nd edn. A K Peters Ltd, WellesleyGoogle Scholar
  2. Bouton CL (1902) Nim, a game with a complete mathematical theory. Ann Math 3:35–39CrossRefGoogle Scholar
  3. Brouwer AE The game of Chomp,
  4. Brouwer AE, Christensen JD Counterexamples to Conjectures About Subset Takeaway and Counting Linear Extensions of a Boolean Lattice. ArXiv preprint arXiv:1702.03018 [math.CO]
  5. Brouwer AE, Cohen AM, Neumaier A (1989) Distance-regular graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 18. Springer-Verlag, Berlin CrossRefGoogle Scholar
  6. Christensen JD, Tilford M (1997) David Gale’s subset take-away game. Amer Math Mon 104:762–766CrossRefGoogle Scholar
  7. Draisma J, van Rijnswou S (2002) How to chomp forests, and some other graphs. Preprint.
  8. Fenner SA, Rogers J (2015) Combinatorial game complexity: an introduction with poset games. Bull Eur Assoc Theor Comput Sci EATCS 116:42–75Google Scholar
  9. Gale D (1974) A curious Nim-type game. Amer Math Mon 81:876–879CrossRefGoogle Scholar
  10. Gale D, Neyman A (1982) Nim-type games, internat. J Game Theory 11:17–20CrossRefGoogle Scholar
  11. García-Marco I, Knauer K (2017) Chomp on numerical semigroups. Accepted in algebraic combinatorics. Arxiv preprint ArXiv:1705.11034 [math.CO]
  12. Grundy PM (1939) Mathematics and games. Eureka 2:6–8Google Scholar
  13. Jones GA (2005) Automorphisms and regular embeddings of merged Johnson graphs. Euro J Comb 26:417–435CrossRefGoogle Scholar
  14. Khandhawit T, Ye L (2011) Chomp on graphs and subsets. Arxiv prerpint arXiv:1101.2718 [math.CO]
  15. Kummer E (1852) Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J für Die Reine und Angew Math 44:93–146Google Scholar
  16. Lucas E (1878) Théorie des Fonctions Numériques Simplement Périodiques. American Journal of Mathematics 1(2): 184–196, (3):197– 240, (4):289–321 CrossRefGoogle Scholar
  17. O’Sullivan C (2017) A vertex and edge deletion game on graphs. Arxiv preprint arXiv:1709.01354 [math.CO]
  18. Schuh F (1952) Spel van delers. Nieuw Tijdschrift voor Wiskunde 39:299–304Google Scholar
  19. Sprague RP (1935) Über mathematische Kampfspiele. Tohoku Math J 41:438–444Google Scholar
  20. Schwalbe U, Walker P (2001) Zermelo and the early history of game theory. Games Econ Behav 34(1):123–137CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad de La LagunaLa LagunaSpain
  2. 2.Aix Marseille UnivUniversité de Toulon, CNRS, LISMarseilleFrance
  3. 3.CONACYT Research Fellow—Centro de Investigación en MatemáticasGuanajuatoMexico

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