Automorphisms of the Cube \(n^d\)

  • Pavel DvořákEmail author
  • Tomáš Valla
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)


Consider a hypergraph \(H_n^d\) where the vertices are points of the d-dimensional combinatorial cube \(n^d\) and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of \(H_n^d\), i.e., permutations of points of \(n^d\) preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of \(H_n^d\) there exists an automorphism of \(H_n^d\) preserving the colors. We show that this problem is \(\mathsf {GI}\)-complete.


  1. 1.
    Allis, L.V.: Searching for solutions in games and artificial intelligence. Ph.D. thesis, University of Limburg (1994)Google Scholar
  2. 2.
    Arvind, V., Das, B., Köbler, J., Toda, S.: Colored hypergraph isomorphism is fixed parameter tractable. Algorithmica 71, 120–138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arvind, V., Köbler, J.: On hypergraph and graph isomorphism with bounded color classes. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 384–395. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Ballester-Bolinches, A., Esteba-Romero, R., Asaad, M.: Products of Finite Groups. (Walter) De Gruyter, Berlin (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Beck, J.: Tic-Tac-Toe Theory. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  6. 6.
    Booth, K.S., Colbourn, C.J.: Problems polynomially equivalent to graph isomorphism. Technical report CS-77/04, Department of Computer Science, University of Waterloo (1977)Google Scholar
  7. 7.
    Choudum, S.A., Sunitha, V.: Automorphisms of augmented cubes. Int. J. Comput. Math. 85, 1621–1627 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hales, A.W., Jewett, R.: Regularity and positional games. Trans. Am. Math. Soc. 106, 222–229 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Harary, F.: The automorphism group of a hypercube. J. Univers. Comput. Sci. 6, 136–138 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Patashnik, O.: Qubic: \(4 \times 4 \times 4 \) tic-tac-toe. Math. Mag. 53, 202–216 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Silver, R.: The group of automorphisms of the game of 3-dimensional ticktacktoe. Am. Math. Mon. 74, 247–254 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  2. 2.Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic

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