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Antipodal Pairs and Crossing Numbers

Author Stefan Felsner

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LIPIcs.GD.2026.22.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
Keywords
  • graph drawing
  • complete graph
  • crossing number
  • antipodal pair
  • Hill conjecture

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Abstract

This paper makes contributions to the theory of crossing numbers of complete and almost complete graphs. An antipodal pair in a drawing is a pair of vertices such that the stars of the two vertices induce a plane subgraph of the drawing. An optimal drawing of a graph is a drawing with a minimum number of crossings. Hill conjectured that the crossing number of the complete graph is: cr(K_n) = H(n) = 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)2⌋ ⌊(n-3)/2⌋. We establish connections between antipodal pairs and optimal drawings of complete graphs. In particular we show that Hill’s conjecture is true within the class of antipodally shellable drawings.
The graph M_{n}^t is obtained from the complete graph K_n by removing the edges of a matching of size t. For these graphs we conjecture that for large enough n optimal drawings have antipodal pairs. If true this conjecture implies that a conjecture of Mohar about the crossing number of M_{2k}^t and a conjecture of the author about the crossing number of M_{2k+1}^t can be verified by checking "small" cases only.
We also construct a new family of Hill drawings, i.e., a family of drawings of K_n with crossing number equal to H(n). As a consequence of this new construction we obtain that every spherical drawing of K_k is a subdrawing of a Hill drawing of K_n for all n ≥ 2k+1.

Cite As Get BibTex

Stefan Felsner. Antipodal Pairs and Crossing Numbers. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.22

Author Details

Stefan Felsner
  • Technische Universität Berlin, Germany

Acknowledgements

I thank Alexandra Wesolek, Yan Alves Radtke, and Joachim Orthaber for inspiring discussions on the topic.

References

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