Abstract
A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of \(K_n\) with no empty lens. In this setting we prove an upper bound of \(3((n-4)!)\) on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite and upper bounded by n!.
This research started at the 3rd Workshop within the collaborative DACH project Arrangements and Drawings, August 19–23, 2019, in Wergenstein (GR), Switzerland, supported by the German Research Foundation (DFG), the Austrian Science Fund (FWF), and the Swiss National Science Foundation (SNSF). We thank the participants for stimulating discussions. S.F. is supported by DFG Project FE 340/12-1. M.H. is supported by SNSF Project 200021E-171681. K.K. is supported by DFG Project MU 3501/3-1 and within the Research Training Group GRK 2434 Facets of Complexity. I.P. was supported by FWF project I 3340-N35.
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Notes
- 1.
That is, disjoint from \(e_i\) except for possibly a shared endpoint.
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Felsner, S., Hoffmann, M., Knorr, K., Parada, I. (2020). On the Maximum Number of Crossings in Star-Simple Drawings of \(K_n\) with No Empty Lens. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_30
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DOI: https://doi.org/10.1007/978-3-030-68766-3_30
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