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The Hamilton Compression of Highly Symmetric Graphs

Authors Petr Gregor , Arturo Merino , Torsten Mütze



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Author Details

Petr Gregor
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Arturo Merino
  • Department of Mathematics, TU Berlin, Germany
Torsten Mütze
  • Department of Computer Science, University of Warwick, United Kingdom
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic

Acknowledgements

We thank Fedor Petrov and Michal Koucký for ideas that simplified some proofs in the full version. We also thank the reviewers for their helpful comments.

Cite AsGet BibTex

Petr Gregor, Arturo Merino, and Torsten Mütze. The Hamilton Compression of Highly Symmetric Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.54

Abstract

We say that a Hamilton cycle C = (x₁,…,x_n) in a graph G is k-symmetric, if the mapping x_i ↦ x_{i+n/k} for all i = 1,…,n, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x₁,…,x_n equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360^∘/k wedge of the drawing. We refer to the maximum k for which there exists a k-symmetric Hamilton cycle in G as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The cycles we construct have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
Keywords
  • Hamilton cycle
  • Gray code
  • hypercube
  • permutahedron
  • Johnson graph
  • Cayley graph
  • abelian group
  • vertex-transitive

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