Proper Coloring of Geometric Hypergraphs

We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions.

[1]  Jean Cardinal,et al.  Coloring Geometric Range Spaces , 2009, Discret. Comput. Geom..

[2]  Matt Gibson,et al.  Decomposing Coverings and the Planar Sensor Cover Problem , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[3]  K. S. Poh.  On the linear vertex-arboricity of a planar graph , 1990, J. Graph Theory.

[4]  Wayne Goddard,et al.  Acyclic colorings of planar graphs , 1991, Discret. Math..

[5]  János Pach,et al.  Decomposition of multiple coverings into many parts , 2007, SCG '07.

[6]  Balázs Keszegh,et al.  Octants are Cover Decomposable , 2011, Electron. Notes Discret. Math..

[7]  Jean Cardinal,et al.  Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles , 2013, WADS.

[8]  Lihong Ma,et al.  Bisectors and Voronoi Diagrams for Convex Distance Functions , 2000 .

[9]  A. Hales,et al.  Regularity and Positional Games , 1963 .

[10]  Balázs Keszegh,et al.  Coloring Points with Respect to Squares , 2016, Symposium on Computational Geometry.

[11]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[12]  Balázs Keszegh,et al.  Convex Polygons are Self-Coverable , 2014, Discret. Comput. Geom..

[13]  Jean Cardinal,et al.  Coloring planar homothets and three-dimensional hypergraphs , 2011, Comput. Geom..

[14]  Franziska Hoffmann,et al.  Spatial Tessellations Concepts And Applications Of Voronoi Diagrams , 2016 .

[15]  János Pach,et al.  Unsplittable Coverings in the Plane , 2013, WG.

[16]  Géza Tóth,et al.  Convex Polygons are Cover-Decomposable , 2010, Discret. Comput. Geom..

[17]  Dömötör Pálvölgyi,et al.  Decomposition of Geometric Set Systems and Graphs , 2010, ArXiv.

[18]  János Pach,et al.  Indecomposable Coverings , 2005, Canadian Mathematical Bulletin.

[19]  Jean Cardinal,et al.  Making Triangles Colorful , 2013, J. Comput. Geom..

[20]  Michiel H. M. Smid,et al.  On the Stretch Factor of Convex Delaunay Graphs , 2008, ISAAC.

[21]  Balázs Keszegh,et al.  Octants are cover-decomposable into many coverings , 2014, Comput. Geom..

[22]  Shakhar Smorodinsky,et al.  On The Chromatic Number of Geometric Hypergraphs , 2007, SIAM J. Discret. Math..

[23]  Dömötör Pálvölgyi,et al.  Indecomposable Coverings with Concave Polygons , 2010, Discret. Comput. Geom..

[24]  Balázs Keszegh,et al.  Online and Quasi-online Colorings of Wedges and Intervals , 2016, Order.

[25]  Arnau Padrol,et al.  Neighborly inscribed polytopes and delaunay triangulations , 2013, ArXiv.

[26]  János Pach,et al.  Covering the plane with convex polygons , 1986, Discret. Comput. Geom..

[27]  János Pach,et al.  Coloring axis-parallel rectangles , 2010, J. Comb. Theory, Ser. A.

[28]  Gwenaël Joret,et al.  Colouring Planar Graphs With Three Colours and No Large Monochromatic Components , 2013, Combinatorics, Probability and Computing.

[29]  Radoslav Fulek.  Coloring geometric hypergraph defined by an arrangement of half-planes , 2010, CCCG.

[30]  Gábor Tardos,et al.  Multiple Coverings of the Plane with Triangles , 2007, Discret. Comput. Geom..

[31]  Jean Cardinal,et al.  Making Octants Colorful and Related Covering Decomposition Problems , 2014, SIAM J. Discret. Math..

[32]  István Kovács.  Indecomposable Coverings with Homothetic Polygons , 2015, Discret. Comput. Geom..

[33]  János Pach,et al.  Delaunay graphs of point sets in the plane with respect to axis‐parallel rectangles , 2008, SODA '08.

[34]  Balázs Keszegh,et al.  An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes , 2014, WG.

[35]  Rajeev Motwani,et al.  Storage management for evolving databases , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[36]  Balázs Keszegh,et al.  More on decomposing coverings by octants , 2015, J. Comput. Geom..

[37]  J. Pach.  Decomposition of multiple packing and covering , 1980 .

[38]  Balázs Keszegh.  Coloring half-planes and bottomless rectangles , 2012, Comput. Geom..

[39]  J. Pach,et al.  Survey on Decomposition of Multiple Coverings , 2013 .

[40]  Shakhar Smorodinsky,et al.  Polychromatic coloring for half-planes , 2012, J. Comb. Theory, Ser. A.