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Harborth’s Conjecture for 4-Regular Planar Graphs

Authors Daniel J. Chang, Timothy Sun

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Author Details

Daniel J. Chang
  • Department of Computer Science, San Francisco State University, CA, USA
Timothy Sun
  • Department of Computer Science, San Francisco State University, CA, USA

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Daniel J. Chang and Timothy Sun. Harborth’s Conjecture for 4-Regular Planar Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 38:1-38:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.38

Abstract

We show that every 4-regular planar graph has a straight-line embedding in the plane where all edges have integer length. The construction extends earlier ideas for finding such embeddings for 4-regular planar graphs with diamond subgraphs or small edge cuts.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Planar graph
  • straight-line embedding
  • Diophantine equation

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References

  1. Vladimir I. Benediktovich. On rational approximation of a geometric graph. Discrete Mathematics, 313(20):2061-2064, 2013. URL: https://doi.org/10.1016/J.DISC.2013.06.018.
  2. T. G. Berry. Points at rational distance from the vertices of a triangle. Acta Arithmetica, 62:391-398, 1992. Google Scholar
  3. Therese Biedl. Drawing some planar graphs with integer edge-lengths. In Canadian Conference on Computational Geometry, pages 291-296, 2011. Google Scholar
  4. Reinhard Diestel. Graph Theory, volume 173. Springer, 2017. Google Scholar
  5. Artūras Dubickas. On some rational triangles. Mediterranean Journal of Mathematics, 9(1):95-103, 2012. Google Scholar
  6. István Fáry. On straight-line representation of planar graphs. Acta Sci. Math. (Szeged), 11:229-233, 1948. Google Scholar
  7. Jim Geelen, Anjie Guo, and David McKinnon. Straight line embeddings of cubic planar graphs with integer edge lengths. Journal of Graph Theory, 58(3):270-274, 2008. URL: https://doi.org/10.1002/JGT.20304.
  8. Arnfried Kemnitz and Heiko Harborth. Plane integral drawings of planar graphs. Discrete Mathematics, 236:191-195, 2001. URL: https://doi.org/10.1016/S0012-365X(00)00442-8.
  9. Henri Lebesgue. Quelques conséquences simples de la formule d'Euler. Journal de Mathématiques Pures et Appliquées, 19(1-4):27-43, 1940. Google Scholar
  10. Tamara Mchedlidze, Martin Nöllenburg, and Ignaz Rutter. Drawing planar graphs with a prescribed inner face. In International Symposium on Graph Drawing, pages 316-327. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03841-4_28.
  11. Anthony Nixon and John Owen. An inductive construction of (2, 1)-tight graphs. arXiv preprint arXiv:1103.2967, 2011. Google Scholar
  12. Sherman K. Stein. Convex maps. Proceedings of the American Mathematical Society, 2(3):464-466, 1951. Google Scholar
  13. Timothy Sun. Rigidity-theoretic constructions of integral Fary embeddings. In Canadian Conference on Computational Geometry, pages 287-290, 2011. Google Scholar
  14. Timothy Sun. Drawing some 4-regular planar graphs with integer edge lengths. In Canadian Conference on Computational Geometry, pages 193-198, 2013. Google Scholar
  15. Klaus Wagner. Bemerkungen zum Vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, 46:26-32, 1936. Google Scholar
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