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The Crossing Number of Twisted Graphs

Abstract

We consider twisted graphs, that is, topological graphs that are weakly isomorphic to subgraphs of the complete twisted graph \(T_n.\) We determine the exact minimum number of crossings of edges among the set of twisted graphs with n vertices and m edges; state a version of the crossing lemma for twisted graphs and conclude that the mid-range crossing constant for twisted graphs is 1/6. Let \(e_k(n)\) be the maximum number of edges over all twisted graphs with n vertices and local crossing number at most k. We give lower and upper bounds for \(e_k(n)\) and settle its exact value for \(k\in \{0,1,2,3,6,10\}.\) We conjecture that for every \(t\ge 1,\) \(e_{{( {\begin{array}{*{20}c} t \\ 2 \\ \end{array} })}} (n) = (t + 1)n - \left( {\begin{array}{*{20}c} {t + 2} \\ 2 \\ \end{array} } \right),\) \(n \ge t+1.\)

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

The authors sincerely thank the referees for helpful comments and corrections on the manuscript. Ana Paulina Figueroa, Juan José Montellano-Ballesteros and Eduardo Rivera-Campo were partially supported by CONACyT-México project A1-S-12891. Juan José Montellano Ballesteros was also supported by PAPIIT-México project IN108121.

Funding

Authors Ana Paulina Figueroa, Juan José Montellano-Ballesteros and Eduardo Rivera-Campo were partially supported by CONACyT-México project A1-S-12891. Juan José Montellano Ballesteros was also supported by PAPIIT-México project IN108121.

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Correspondence to Silvia Fernández-Merchant.

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Ábrego, B.M., Fernández-Merchant, S., Figueroa, A.P. et al. The Crossing Number of Twisted Graphs. Graphs and Combinatorics 38, 134 (2022). https://doi.org/10.1007/s00373-022-02538-3

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  • DOI: https://doi.org/10.1007/s00373-022-02538-3

Keywords

  • Crossing number
  • Twisted graph
  • Twisted drawing
  • Crossing lemma
  • Mid-range crossing constant

Mathematics Subject Classification

  • 05C10
  • 05C35
  • 05C62
  • 52C10
  • 68R10