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Crossing Numbers and Combinatorial Characterization of Monotone Drawings of \(K_n\)

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Abstract

In 1958, Hill conjectured that the minimum number of crossings in a drawing of \(K_n\) is exactly \(Z(n) = \frac{1}{4} \big \lfloor \frac{n}{2}\big \rfloor \big \lfloor \frac{n-1}{2}\big \rfloor \big \lfloor \frac{n-2}{2}\big \rfloor \big \lfloor \frac{n-3}{2}\big \rfloor \). Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by \(x\)-monotone curves. In fact, our proof shows that the conjecture remains true for \(x\)-monotone drawings of \(K_n\) in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by Ábrego et al. We also give a combinatorial characterization of several classes of \(x\)-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Carathéodory’s theorem to simple drawings of complete graphs.

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Acknowledgments

The authors were supported by the grant GAČR GIG/11/E023 GraDR in the framework of ESF EUROGIGA program. The first and the third author were also supported by the Grant Agency of the Charles University, GAUK 1262213, and by the grant SVV-2013-267313 (Discrete Models and Algorithms). The third author was also partially supported by ERC Advanced Research Grant no 267165 (DISCONV). The second author gratefully acknowledges support from the Swiss National Science Foundation Grant PBELP2_146705. We would like to thank Pavel Valtr for initializing the research which led to this problem and Marek Eliáš for developing visualization tools that were helpful during the research.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Prague 1, Czech Republic

    Martin Balko & Radoslav Fulek

  2. Department of Applied Mathematics and Institute for Theoretical Computer Science, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Prague 1, Czech Republic

    Jan Kynčl

  3. IEOR, Columbia University, New York, NY, USA

    Radoslav Fulek

  4. Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Jan Kynčl

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  1. Martin Balko
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Correspondence to Jan Kynčl.

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Balko, M., Fulek, R. & Kynčl, J. Crossing Numbers and Combinatorial Characterization of Monotone Drawings of \(K_n\) . Discrete Comput Geom 53, 107–143 (2015). https://doi.org/10.1007/s00454-014-9644-z

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