Coloring Circle Arrangements: New 4-Chromatic Planar Graphs

Man Kwun Chiu, Stefan Felsner, Manfred Scheucher*, Felix Schröder, Raphael Steiner, Birgit Vogtenhuber

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. This paper is motivated by the conjecture. We show that the conjecture holds in the special case when the arrangement is ▵ -saturated, i.e., arrangements where one color class of the 2-coloring of faces consists of triangles only. Moreover, we extend ▵ -saturated arrangements with certain properties to a family of arrangements which are 4-chromatic. The construction has similarities with Koester’s (1985) crowning construction. We also investigate fractional colorings. We show that every arrangement A of pairwise intersecting pseudocircles is “close” to being 3-colorable; more precisely χf(A)≤3+O(1n) where n is the number of pseudocircles. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves the conjecture of Gimbel, Kündgen, Li and Thomassen (2019) that every 4-chromatic planar graph has fractional chromatic number strictly greater than 3.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages8
Publication statusPublished - 2021

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • Arrangement of pseudolines and pseudocircles
  • Chromatic number
  • Critical graph
  • Fractional coloring
  • Triangle-saturated

ASJC Scopus subject areas

  • Mathematics(all)


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