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Generation of Cycle Permutation Graphs and Permutation Snarks

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a 2-factor consisting of two chordless cycles, and non-hamiltonian cycle permutation graphs, from which the permutation snarks can easily be computed. This allows us to generate all cycle permutation graphs up to order 34 and all permutation snarks up to order 46, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order \(6\bmod 8\) or a smallest permutation snark of girth at least 6. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to a conjecture by Zhang.

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Acknowledgements

The authors thank Steven Van Overberghe for suggesting the idea of Algorithm 2 and Edita Mácajová and Martin Škoviera for their valuable insights and contributions.

This research was supported by Internal Funds of KU Leuven and by an FWO grant with grant number G0AGX24N. Several of the computations for this work were carried out using the supercomputer infrastructure provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government.

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Authors and Affiliations

  1. Department of Computer Science, KU Leuven Kulak, 8500, Kortrijk, Belgium

    Jan Goedgebeur & Jarne Renders

  2. Department of Applied Mathematics, Computer Science and Statistics, Ghent University, 9000, Ghent, Belgium

    Jan Goedgebeur

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  1. Jan Goedgebeur
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  2. Jarne Renders
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Correspondence to Jarne Renders .

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Editors and Affiliations

  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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Goedgebeur, J., Renders, J. (2025). Generation of Cycle Permutation Graphs and Permutation Snarks. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_24

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  • DOI: https://doi.org/10.1007/978-3-031-82670-2_24

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