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International Computing and Combinatorics Conference

COCOON 2016: Computing and Combinatorics pp 405–416Cite as

Automorphisms of the Cube \(n^d\)

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9797))

Abstract

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.

P. Dvořák—The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 616787.

T. Valla—Supported by the Centre of Excellence – Inst. for Theor. Comp. Sci. 79 (project P202/12/G061 of GA ČR).

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Notes

  1. 1.

    The parameter is the maximum number of vertices colored by the same color.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic

    Pavel Dvořák

  2. Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Valla

Authors
  1. Pavel Dvořák
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  2. Tomáš Valla
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Corresponding author

Correspondence to Pavel Dvořák .

Editor information

Editors and Affiliations

  1. Virginia Commonwealth Univ , Richmond, Virginia, USA

    Thang N. Dinh

  2. University of Florida , Gainesville, Alabama, USA

    My T. Thai

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© 2016 Springer International Publishing Switzerland

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Dvořák, P., Valla, T. (2016). Automorphisms of the Cube \(n^d\) . In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_33

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  • DOI: https://doi.org/10.1007/978-3-319-42634-1_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42633-4

  • Online ISBN: 978-3-319-42634-1

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