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Erschienen in:
01.02.2016
Crossing edges and faces of line arrangements in the plane
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 2/2016
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Abstract
For any natural number \(n\) we define \(f(n)\) to be the minimum number with the following property. Given any arrangement \(\mathcal{A}(\mathcal{L})\) of \(n\) blue lines in the real projective plane one can find \(f(n)\) red lines different from the blue lines such that any edge in the arrangement \(\mathcal{A}(\mathcal{L})\) is crossed by a red line. We define \(h(n)\) to be the minimum number with the following property. Given any arrangement \(\mathcal{A}(\mathcal{L})\) of \(n\) blue lines in the real projective plane one can find \(h(n)\) red lines different from the blue lines such that every face in the arrangement \(\mathcal{A}(\mathcal{L})\) is crossed in its interior by a red line. In this paper we show \(f(n)=2n-o(n)\) and \(h(n)=n-o(n)\).