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The Crossing Number of Twisted Graphs

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

References

  1. Ábrego, B.M., Aichholzer, O., Fernández-Merchant, S., Ramos, P., Salazar, G.: Shellable drawings and the cylindrical crossing number of \(K_n\). Discrete Comput. Geom. 52–4, 743–753 (2014). https://doi.org/10.1007/s00454-014-9635-0

    Article  MATH  Google Scholar 

  2. Ábrego, B.M., Aichholzer, O., Fernández-Merchant, S., Ramos, P., Salazar, G.: The 2-page crossing number of \(K_n\). Discrete Comput. Geom. 49–4, 747–777 (2013). https://doi.org/10.1007/s00454-013-9514-0

    Article  MATH  Google Scholar 

  3. Ábrego, B.M., Aichholzer, O., Dandurand, J., Fernández-Merchant, S., Lagoda, E., Sapozhnikov, Y.: Improved lower bounds on book crossing numbers of \(K_n\). In: Proceedings JCDCG3, Tokyo University of Science (2016)

  4. Ábrego, B.M., Fernández-Merchant, S.: The crossing lemma for convex graphs. Preprint

  5. Ackerman, E.: On topological graphs with at most four crossings per edge. Comput. Geom. 85, 101574 (2019). https://doi.org/10.1016/j.comgeo.2019.101574

    Article  MathSciNet  MATH  Google Scholar 

  6. Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, A.: Crossing-free subgraphs. Ann. Discrete Math. 12, 9–12 (1982). https://doi.org/10.1016/S0304-0208(08)73484-4

    Article  MathSciNet  MATH  Google Scholar 

  7. Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Algebraic Discrete Methods 8–1, 33–58 (1987). https://doi.org/10.1137/0608002

    Article  MathSciNet  MATH  Google Scholar 

  8. Guy, R.K.: The decline and fall of Zarankiewicz’s theorem. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 63–69. Academic Press, New York (1969)

    Google Scholar 

  9. Harary, F., Hill, H.: On the number of crossings in a complete graph. Proc. Edinb. Math. Soc. 13–4, 333–338 (1963). https://doi.org/10.1017/S0013091500025645

    Article  MathSciNet  MATH  Google Scholar 

  10. Harborth, H., Mengersen, I.: Drawings of the complete graph with maximum number of crossings. In: Proceedings of the 23rd Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. 88, pp. 225–228. Utilitas Mathematica, Winnipeg (1992)

  11. Leighton, T.: Complexity Issues in VLSI, Foundations of Computing Series. MIT Press, Cambridge (1983)

    Google Scholar 

  12. Omaña-Pulido, E., Rivera-Campo, E.: Notes on the twisted graph. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol. 7579, pp. 119–125. Springer, Berlin (2012)

  13. Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom. 36, 527–552 (2006). https://doi.org/10.1007/s00454-006-1264-9

  14. Pach, J., Solymosi, J., Tóth, G.: Unavoidable configurations in complete topological graphs. Discrete Comput. Geom. 30, 311–320 (2003). https://doi.org/10.1007/s00454-003-0012-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Pach, J., Spencer, J., Tóth, G.: New bounds on crossing numbers. Discrete Comput. Geom. 24, 623–644 (2000). https://doi.org/10.1007/s004540010011

    Article  MathSciNet  MATH  Google Scholar 

  16. Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17–3, 427–439 (1997). https://doi.org/10.1007/BF01215922

    Article  MathSciNet  MATH  Google Scholar 

  17. Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: Proceedings of the Symposium on Graph Drawing (GD’97). LNCS, vol. 1353, pp. 248–261. Springer, Berlin (1997). https://doi.org/10.1007/3-540-63938-1_67

  18. Purchase, H.C., Carrington, D., Allder, J.: Graph layout aesthetics in UML diagrams: user preferences. J. Graph Algorithms Appl. 6–3, 255–279 (2002). https://doi.org/10.1007/3-540-44541-2_2

    Article  MathSciNet  MATH  Google Scholar 

  19. Schaefer, M.: Crossing Numbers of Graphs. CRC Press, Boca Raton (2018). https://doi.org/10.1201/9781315152394

    Book  MATH  Google Scholar 

  20. Schaefer, M.: The graph crossing number and its variants: a survey. Electron. J. Comb. DS21 (2022). https://doi.org/10.37236/2713

  21. Székely, L.A.: Crossing numbers and hard Erdos problems in discrete geometry. Probab. Comput. 7, 353–358 (1997). https://doi.org/10.1017/S0963548397002976

    Article  MathSciNet  MATH  Google Scholar 

  22. Turán, P.: A note of welcome. J. Graph Theory 1, 7–9 (1977). https://doi.org/10.1002/jgt.3190010105

    Article  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematics, CSUN, Northridge, CA, USA

    Bernardo M. Ábrego & Silvia Fernández-Merchant

  2. Departamento Académico de Matemáticas, ITAM, Mexico City, Mexico

    Ana Paulina Figueroa

  3. Instituto de Matemáticas, UNAM, Mexico City, Mexico

    Juan José Montellano-Ballesteros

  4. Departamento de Matemáticas, UAM-Iztapalapa, Mexico City, Mexico

    Eduardo Rivera-Campo

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  1. Bernardo M. Ábrego
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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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