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DOI: 10.4230/LIPIcs.MFCS.2025.15

Symmetry Classes of Hamiltonian Cycles

Authors: Júlia Baligács, Sofia Brenner, Annette Lutz, and Lena Volk

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.

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Júlia Baligács, Sofia Brenner, Annette Lutz, and Lena Volk. Symmetry Classes of Hamiltonian Cycles. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{baligacs_et_al:LIPIcs.MFCS.2025.15,
  author =	{Balig\'{a}cs, J\'{u}lia and Brenner, Sofia and Lutz, Annette and Volk, Lena},
  title =	{{Symmetry Classes of Hamiltonian Cycles}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.15},
  URN =		{urn:nbn:de:0030-drops-241221},
  doi =		{10.4230/LIPIcs.MFCS.2025.15},
  annote =	{Keywords: Hamiltonian cycles, graph automorphisms, Cayley graphs, abelian groups, Cartesian product of graphs}
}

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Symmetry Classes of Hamiltonian Cycles

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