Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs

EYAL ACKERMAN, BALÁZS KESZEGH, DÖMÖTÖR PÁLVÖLGYI

Research output: Contribution to journalArticlepeer-review

Abstract

What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.

Original languageEnglish
Pages (from-to)2250-2269
Number of pages20
JournalSIAM Journal on Discrete Mathematics
Volume34
Issue number4
DOIs
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Keywords

  • Coloring
  • Combinatorial geometry
  • Geometric hypergraph
  • Pseudoline

ASJC Scopus subject areas

  • General Mathematics

Cite this