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Chomp on generalized Kneser graphs and others

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Abstract

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.

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Acknowledgements

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.

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Correspondence to Luis Pedro Montejano.

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L.P.M. has been supported by the Mexican National Council on Science and Technology (Cátedras-CONACYT). K.K. has been supported by ANR projects GATO: ANR-16-CE40-0009-01, DISTANCIA: ANR-17-CE40-0015, and CAPPS: ANR-17-CE40-0018. I. G. has been supported by Ministerio de Economía y Competitividad, Spain (MTM2016-78881-P). This research was initiated on a visit of I.G. and K.K. at CIMAT supported by UMI Laboratoire Solomon Leftschetz— LaSol—no. 2001 CNRS-CONACYT-UNAM, Mexique.

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García-Marco, I., Knauer, K. & Montejano, L.P. Chomp on generalized Kneser graphs and others. Int J Game Theory (2019). https://doi.org/10.1007/s00182-019-00697-x

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Keywords

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

Mathematics Subject Classification

  • 05C57
  • 91A05
  • 91A46
  • 91A43
  • 91A05