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

Abstract

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
Pages84-91
Number of pages8
DOIs
Publication statusPublished - 2021

Publication series

NameTrends in Mathematics
Volume14
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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