Coloring Jordan regions and curves

Wouter Cames Van Batenburg, Louis Esperet, Tobias Müller

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family F of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most κ elements of F (with κ sufficiently large), then we show that the elements of F can be colored with at most κ + 1 colors so that intersecting Jordan regions are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a problem of Hliněný (1998) on the chromatic number of contact systems of strings. We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.

Original languageEnglish
Pages (from-to)1670-1684
Number of pages15
JournalSIAM Journal on Discrete Mathematics
Volume31
Issue number3
DOIs
Publication statusPublished - 2017

Keywords

  • Geometric graphs
  • Graph coloring
  • Jordan curves
  • Jordan regions

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