On the 2-colored crossing number

Oswin Aichholzer; Ruy Fabila-Monroy; Adrian Fuchs; Carlos Hidalgo-Toscano; Irene Parada; Birgit Vogtenhuber; Francisco Zaragoza

arXiv · 1908.06461

On the 2-colored crossing number

Oswin Aichholzer·Ruy Fabila-Monroy·Adrian Fuchs·Carlos Hidalgo-Toscano·Irene Parada·Birgit Vogtenhuber·Francisco Zaragoza

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Abstract

Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$. To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices. We obtain such instances using a combination of heuristics and integer programming. Second, for any fixed drawing of $K_n$, we improve the bound on the ratio between its rectilinear 2-colored crossing number and its rectilinear crossing number.

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Oswin Aichholzer, Ruy Fabila-Monroy, Adrian Fuchs, Carlos Hidalgo-Toscano, Irene Parada, Birgit Vogtenhuber, Francisco Zaragoza. 2019-08-18. On the 2-colored crossing number. https://arxiv.org/abs/1908.06461

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