Abstract
We consider some problems on red and blue points in the plane lattice. An L-line segment in the plane lattice consists of a vertical line segment and a horizontal line segment having a common endpoint. There are some results on geometric graphs on a set of red and blue points in the plane. We show that some similar results also hold for a set of red and blue points in the plane lattice using L-line segments instead of line segments. For example, we show that if n red points and n blue points are given in the plane lattice in general position, then there exists a noncrossing geometric perfect matching covering them, each of whose edges is an L-line segment and connects a red point and a blue point.
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Notes
- 1.
In the plane, no three points lie on the same line if and only if every three points make a triangle. Similarly, in the plane lattice, every vertical line and horizontal line pass through at most one point if and only if every two points make a digon with two L-line segments.
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Kano, M., Suzuki, K. (2013). Discrete Geometry on Red and Blue Points in the Plane Lattice. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_18
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DOI: https://doi.org/10.1007/978-1-4614-0110-0_18
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