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arXiv · 1610.05498

Automorphisms of the Cube $n^d$

Pavel Dvořák·Tomáš Valla
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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 ${\sf GI}$-complete.

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Pavel Dvořák, Tomáš Valla. 2016-10-18. Automorphisms of the Cube $n^d$. https://doi.org/10.1016/j.disc.2020.112234

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