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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
References
I. Bárány, J. Matoušek, Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25, 317–334 (2001)
S. Bereg, Orthogonal equipartitions. Comput. Geom. 42, 305–314 (2009)
A. Kaneko, M. Kano, A balanced interval of two sets of points on a line. Combinatorial Geometry and Graph Theory. LNCS, vol. 3330 (Springer, Berlin, 2005), pp. 108–112
M. Uno, T. Kawano, M. Kano, Bisections of two sets of points in the plane lattice, IEICE Transactions on Fundamentals vol. E92-A, 502–507 (2009)
M. Kano, K. Suzuki, Geometric Graphs in the Plane Lattice, XIV Spanish meeting on computational geometry, LNCS, Springer-Verlag, Berlin,7579, 204–281 (2012)
S. Tokunaga, Intersection number of two connected geometric graphs. Inform. Process. Lett. 59, 331–333 (1996)
M. Kano, K. Suzuki, M. Uno, T. Kawano, M. Kano, Bisections of two sets of points in the plane lattice. IEICE Trans. 340, Fundament. E92-A, 502507 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4614-0110-0_18
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-0109-4
Online ISBN: 978-1-4614-0110-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)