Crossing edges and faces of line arrangements in the plane

Rom Pinchasi ()
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Rom Pinchasi: Technion—Israel Institute of Technology

Journal of Combinatorial Optimization, 2016, vol. 31, issue 2, No 6, 533-545

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

Keywords: Line arrangement; Faces; edges (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-014-9769-2

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