Document Open Access Logo

Generalized Snarks, Disjoint Perfect Matchings, and Graph Covers

Authors Filip Filipi , Jan Kratochvíl , Roman Nedela

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2026.45.pdf
  • Filesize: 1.12 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • graph
  • graph cover
  • perfect matching
  • NP-completeness

Metrics

Abstract

We explore the interplay among three classical notions in graph theory: edge-colorings, perfect matchings, and graph coverings (locally bijective homomorphisms of graphs). In this paper, we consider undirected graphs in full generality of this notion: in contrast to the standard notion of a simple graph, our graphs may contain loops, semi-edges, and multiple edges. Many well-studied graph concepts, including matchings, edge-colorings, and covering projections, extend naturally to such graphs. Nevertheless, the role of simple graphs for graph covering problems is central, as emphasized in [J. Bok, J. Fiala, N. Jedličková, J. Kratochvíl, and M. Seifrtová. Computational complexity of covering disconnected multigraphs. Discret. Appl. Math., 359:229–243, 2024]. In that work, a relation "being stronger" was defined (a graph A is stronger than a graph B if every simple graph that covers A also covers B), and it was conjectured that if A has no semi-edges, then A is stronger than B if and only if A covers B. In their extended abstract presented at Eurocomb'23, Kratochvíl and Nedela proved this conjecture for 3-regular 1-vertex graphs B (and arbitrary A). They also introduced the notion (A,B)-snark for a simple graph G that demonstrates that A is not stronger than B. We continue this line of research in the current paper.
As the main result, we show that for every graph A, there exists a simple graph D that covers A in such a way that the maximum number of pairwise disjoint perfect matchings equals the maximum number of pairwise disjoint perfect semi-matchings in A, i.e., spanning 1-regular subgraphs. Notably, the proof is constructive. As a corollary, we obtain a necessary condition for A to be stronger than B in general. This condition turns out to be sufficient whenever B is a 1-vertex graph (there are infinitely many of them), which, in particular, proves the aforementioned conjecture of Bok et al. in this case. Finally, we provide a constructive alternative to the existential NP-hardness proof of covering disconnected graphs in Bok et al. for the case when the target graph contains a 1-vertex component which itself determines an NP-hard covering problem.

Cite As Get BibTex

Filip Filipi, Jan Kratochvíl, and Roman Nedela. Generalized Snarks, Disjoint Perfect Matchings, and Graph Covers. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.45

Author Details

Filip Filipi
  • Institute for Applied Mathematics, University of Bonn, Germany
Jan Kratochvíl
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Roman Nedela
  • Department of Mathematics, Faculty of Applied Sciences, West Bohemia University, Pilsen, Czech Republic

Funding

  • Filipi, Filip: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2047/2 – 390685813.
  • Nedela, Roman: Supported by the grant VEGA 2/0056/25 of the Ministry of Education, Development, Research and Youth of the Slovak Republic.

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments.

References

  1. James Abello, Michael R. Fellows, and John C. Stillwell. On the complexity and combinatorics of covering finite complexes. Australian Journal of Combinatorics, 4:103-112, 1991. URL: http://ajc.maths.uq.edu.au/pdf/4/ocr-ajc-v4-p103.pdf.
  2. Dana Angluin. Local and global properties in networks of processors (extended abstract). In Raymond E. Miller, Seymour Ginsburg, Walter A. Burkhard, and Richard J. Lipton, editors, Proceedings of the 12th Annual ACM Symposium on Theory of Computing, April 28-30, 1980, Los Angeles, California, USA, pages 82-93. ACM, 1980. URL: https://doi.org/10.1145/800141.804655.
  3. Dana Angluin and A. Gardiner. Finite common coverings of pairs of regular graphs. Journal of Combinatorial Theory B, 30(2):184-187, 1981. URL: https://doi.org/10.1016/0095-8956(81)90062-9.
  4. Hans L. Bodlaender. The classification of coverings of processor networks. Journal of Parallel Distributed Computing, 6(1):166-182, 1989. URL: https://doi.org/10.1016/0743-7315(89)90048-8.
  5. Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, and Jan Kratochvíl. Computational complexity of covering multigraphs with semi-edges: Small cases. In Filippo Bonchi and Simon J. Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, Tallinn, Estonia, August 23-27, 2021, volume 202 of LIPIcs, pages 21:1-21:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.21.
  6. Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, and Jan Kratochvíl. Computational complexity of covering multigraphs with semi-edges: Small cases. J. Comput. Syst. Sci., 156:103714, 2026. URL: https://doi.org/10.1016/J.JCSS.2025.103714.
  7. Jan Bok, Jiří Fiala, Nikola Jedličková, Jan Kratochvíl, and Pawel Rzążewski. List covering of regular multigraphs. In Cristina Bazgan and Henning Fernau, editors, Combinatorial Algorithms - 33rd International Workshop, IWOCA 2022, Trier, Germany, June 7-9, 2022, Proceedings, volume 13270 of Lecture Notes in Computer Science, pages 228-242. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06678-8_17.
  8. Jan Bok, Jiří Fiala, Nikola Jedličková, Jan Kratochvíl, and Pawel Rzążewski. List covering of regular multigraphs with semi-edges. Algorithmica, 86(3):782-807, 2024. URL: https://doi.org/10.1007/S00453-023-01163-7.
  9. Jan Bok, Jiří Fiala, Nikola Jedličková, Jan Kratochvíl, and Michaela Seifrtová. Computational complexity of covering disconnected multigraphs. In Evripidis Bampis and Aris Pagourtzis, editors, Fundamentals of Computation Theory - 23rd International Symposium, FCT 2021, Athens, Greece, September 12-15, 2021, Proceedings, volume 12867 of Lecture Notes in Computer Science, pages 85-99. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86593-1_6.
  10. Jan Bok, Jiří Fiala, Nikola Jedličková, Jan Kratochvíl, and Michaela Seifrtová. Computational complexity of covering disconnected multigraphs. Discret. Appl. Math., 359:229-243, 2024. URL: https://doi.org/10.1016/J.DAM.2024.07.035.
  11. Gunnar Brinkmann, Jan Goedgebeur, Jonas Hägglund, and Klas Markström. Generation and properties of snarks. J. Comb. Theory B, 103(4):468-488, 2013. URL: https://doi.org/10.1016/J.JCTB.2013.05.001.
  12. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449-467, 1965. URL: https://doi.org/10.4153/CJM-1965-045-4.
  13. Miquel A. Fiol, Giuseppe Mazzuoccolo, and Eckhard Steffen. Measures of edge-uncolorability of cubic graphs. Electron. J. Comb., 25(4):4, 2018. URL: https://doi.org/10.37236/6848.
  14. Martin Gardner. Mathematical games. Scientific American, 234(4):126-131, 1976. URL: http://www.jstor.org/stable/24950334.
  15. Ezra Getzler and Mikhail M. Kapranov. Modular operads. Compositio Mathematica, 110(1):65-125, 1998. URL: https://doi.org/10.1023/A:1000245600345.
  16. Ian Holyer. The NP-completeness of edge-coloring. SIAM J. Comput., 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  17. Martin Kochol. Snarks without small cycles. J. Comb. Theory B, 67(1):34-47, 1996. URL: https://doi.org/10.1006/JCTB.1996.0032.
  18. Jan Kratochvíl and Roman Nedela. Graph covers and generalized snarks (extended abstract). In Daniel Král and Jaroslav Nešetřil, editors, Proceedings of EUROCOMB'23, pages 681-687. MUNI Press, 2023. URL: https://doi.org/10.5817/CZ.MUNI.EUROCOMB23-094.
  19. Jan Kratochvíl and Roman Nedela. Graph covers and semi-covers: Who is stronger?, 2025. URL: https://doi.org/10.48550/arXiv.2504.17387v1.
  20. Jin Ho Kwak and Roman Nedela. Graphs and their coverings. Lecture Notes Series, 17, 2007. Google Scholar
  21. Daniel Leven and Zvi Galil. NP-completeness of finding the chromatic index of regular graphs. Journal of Algorithms, 4(1):35-44, 1983. URL: https://doi.org/10.1016/0196-6774(83)90032-9.
  22. Yulai Ma, Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf. Edge-connectivity and pairwise disjoint perfect matchings in regular graphs. Comb., 44(2):429-440, 2024. URL: https://doi.org/10.1007/S00493-023-00078-9.
  23. Aleksander Malnič, Dragan Marušič, and Primož Potočnik. Elementary abelian covers of graphs. Journal of Algebraic Combinatorics, 20(1):71-97, 2004. URL: https://doi.org/10.1023/B:JACO.0000047294.42633.25.
  24. Aleksander Malnič, Roman Nedela, and Martin Škoviera. Lifting graph automorphisms by voltage assignments. European Journal of Combinatorics, 21(7):927-947, 2000. URL: https://doi.org/10.1006/eujc.2000.0390.
  25. Edita Máčajová and Martin Škoviera. Superposition of snarks revisited. Eur. J. Comb., 91:103220, 2021. URL: https://doi.org/10.1016/j.ejc.2020.103220.
  26. Roman Nedela and Martin Škoviera. Regular embeddings of canonical double coverings of graphs. Journal of Combinatorial Theory, Series B, 67(2):249-277, 1996. URL: https://doi.org/10.1006/jctb.1996.0044.
  27. Julius Petersen. Die theorie der regulären graphs. Acta Mathematica, 15:193-220, 1891. Google Scholar
  28. Julius Petersen. Sur le théorème de Tait. L'Intermédiaire des Mathématiciens, 5:225-227, 1898. Google Scholar
  29. Eckhard Steffen. 1-factor and cycle covers of cubic graphs. J. Graph Theory, 78(3):195-206, 2015. URL: https://doi.org/10.1002/jgt.21798.
  30. P. G. Tait. 10. remarks on the previous communication. Proceedings of the Royal Society of Edinburgh, 10:729-729, 1880. URL: https://doi.org/10.1017/S0370164600044643.
  31. V. G. Vizing. On an estimate of the chromatic class of a p-graph. Diskret. Analiz., 3:25-30, 1964. Google Scholar
  32. Martin Škoviera and Peter Varša. Deciding whether four perfect matchings can cover the edges of a snark is NP-complete. Theor. Comput. Sci., 988:114374, 2024. URL: https://doi.org/10.1016/j.tcs.2023.114374.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact