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Hunting for Directed 2-Spiders

Authors Grzegorz Gutowski , Gaurav Kucheriya

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
Keywords
  • Oriented and Directed Graphs
  • Extremal Graph Theory
  • Mathematics of Computing
  • Unavoidable Subgraphs

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Abstract

Hons, Klimošová, Mikšaník, Tkadlec, Tyomkyn and the second author proved that, for every integer 𝓁 ≥ 1, every directed graph with minimum out-degree at least 3.23 ⋅ 𝓁 contains a (2,𝓁)-spider (a 1-subdivision of the in-star with 𝓁 leaves) as a subgraph. Hons et al. also conjectured that the bound on the minimum out-degree can be further improved to 2 𝓁. In this note, we confirm this conjecture by showing that every directed graph with minimum out-degree at least 2𝓁 contains a (2, 𝓁)-spider as a subgraph. This result is best possible, as the complete directed graph with 2𝓁 vertices does not contain a (2,𝓁)-spider.

Cite As Get BibTex

Grzegorz Gutowski and Gaurav Kucheriya. Hunting for Directed 2-Spiders. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 21:1-21:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.21

Author Details

Grzegorz Gutowski
  • Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Gaurav Kucheriya
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czechia

Funding

  • Gutowski, Grzegorz: Partially supported by grant no. 2023/49/B/ST6/01738 from National Science Centre, Poland.
  • Kucheriya, Gaurav: Partially supported by GAČR grant 25-17377S and GAUK project 92125.

Acknowledgements

The authors would like to thank the organizers of the 10th Cracow Conference on Graph Theory (2025) which brought the authors together to work on this problem. The authors are also grateful to the organizers of Homonolo 2025, where the final stages of this research was carried out. The authors also thank Mykhaylo Tyomkyn for carefully reading an earlier draft of this paper and for helpful comments.

References

  1. Pierre Aboulker, Nathann Cohen, Frédéric Havet, William Lochet, Phablo F. S. Moura, and Stéphan Thomassé. Subdivisions in digraphs of large out-degree or large dichromatic number. Electronic Journal of Combinatorics, 26(3):#P3.19:1-18, 2019. URL: https://doi.org/10.37236/6521.
  2. Sepehr Assadi, Soheil Behnezhad, Sayan Bhattacharya, Martín Costa, Shay Solomon, and Tianyi Zhang. Vizing’s Theorem in near-linear time. In STOC 2025: 57th Annual ACM Symposium on Theory of Computing, pages 24-35, 2025. URL: https://doi.org/10.1145/3717823.3718265.
  3. Sepehr Assadi, Soheil Behnezhad, Sayan Bhattacharya, Martín Costa, Shay Solomon, and Tianyi Zhang. Vizing’s Theorem in deterministic almost-linear time. In SODA 2026: 37th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 5558-5585, 2026. URL: https://doi.org/10.1137/1.9781611978971.198.
  4. Jørgen Bang-Jensen and Gregory Z. Gutin. Digraphs: Theory, Algorithms and Applications. Springer Science & Business Media, 2008. URL: https://doi.org/10.1007/978-1-84800-998-1.
  5. Louis Caccetta and Roland Häggkvist. On minimal digraphs with given girth. In 9th Southeastern Conference on Combinatorics, Graph Theory and Computing, volume 21 of Congressus Numerantium, pages 181-187, 1978. Google Scholar
  6. Gabriel A. Dirac. Some theorems on abstract graphs. Proceedings of the London Mathematical Society, 2(1):69-81, 1952. URL: https://doi.org/10.1112/plms/s3-2.1.69.
  7. Yahya Ould Hamidoune. A note on minimal directed graphs with given girth. Journal of Combinatorial Theory, Series B, 43(3):343-348, 1987. URL: https://doi.org/10.1016/0095-8956(87)90009-8.
  8. Tomáš Hons, Tereza Klimošová, Gaurav Kucheriya, David Mikšaník, Josef Tkadlec, and Mykhaylo Tyomkyn. Unavoidable subgraphs in digraphs with large out-degrees, 2025. URL: https://arxiv.org/abs/2504.20616.
  9. Wolfgang Mader. Existenz n-fach zusammenhängender teilgraphen in graphen genügend großer kantendichte. Abh. Math. Sem. Universität Hamburg, 37:86-97, 1972. Google Scholar
  10. Wolfgang Mader. Hinreichende bedingungen für die existenz von teilgraphen, die zu einem vollständigen graphen homömorph sind. Math. Nachr., 53(1-6):145-150, 1972. URL: https://doi.org/10.1002/mana.19720530113.
  11. Wolfgang Mader. Degree and local connectivity in digraphs. Combinatorica, 5(2):161-165, 1985. URL: https://doi.org/10.1007/BF02579379.
  12. Øystein Ore. Note on Hamilton circuits. American Mathematical Monthly, 67(1):55-55, 1960. URL: https://doi.org/10.2307%2F2308928.
  13. Blair Dowling Sullivan. A summary of problems and results related to the Caccetta-Haggkvist conjecture, 2006. URL: https://arxiv.org/abs/math/0605646.
  14. Vadim G. Vizing. On an estimate of the chromatic class of a p-graph. Diskretnyi Analiz, 3:25-30, 1964. Google Scholar
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