Colored Ray Configurations

We study the cyclic color sequences induced at infinity by colored rays with apices being a given balanced finite bichromatic point set. We first study the case in which the rays are required to be pairwise disjoint. We derive a lower bound on the number of color sequences that can be realized from any such fixed point set and examine color sequences that can be realized regardless of the point set, exhibiting negative examples as well. We also provide a tight upper bound on the number of configurations that can be realized from a point set, and point sets for which there are asymptotically less configurations than that number. In addition, we provide algorithms to decide whether a color sequence is realizable from a given point set in a line or in general position. We address afterwards the variant of the problem where the rays are allowed to intersect. We prove that for some configurations and point sets, the number of ray crossings must be Θ(n) and study then configurations that can be realized by rays that pairwise cross. We show that there are point sets for which the number of configurations that can be realized by pairwise-crossing rays is asymptotically smaller than the number of configurations realizable by pairwise-disjoint rays. We provide also point sets from which any configuration can be realized by pairwise-crossing rays and show that there is no configuration that can be realized by pairwise-crossing rays from every point set. In memoriam: Ferran Hurtado (1951-2014) This paper is dedicated to the memory of our dear friend and colleague Ferran Hurtado that left us unexpectedly in 2014. ∗Dept. de Matemáticas. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México D.F., México, [email protected] †Dept. de Métodos Estad́ısticos, IUMA, Universidad de Zaragoza, Spain, {olaverri, jtejel}@unizar.es ‡Dept. de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain. §Palma, Spain, [email protected] ¶Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago, Santiago, Chile, [email protected]. Partially supported by projects CONICYT FONDECYT/Regular 1160543 (Chile), and Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 (Chile). ‖Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Pilsen, Czech Republic, [email protected] ∗∗Dept. Matemàtica Aplicada II, Universitat Politècnica de Catalunya, [email protected] ††Instituto de Matemáticas, Universidad Nacional Autónoma de México, México D.F., México, [email protected] 1 ar X iv :1 80 2. 03 41 5v 1 [ cs .C G ] 9 F eb 2 01 8