@inbook{cc691c9571764bffaf9962e1665ca19d,

title = "Coloring Circle Arrangements: New 4-Chromatic Planar Graphs",

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{\textquoteright}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, K{\"u}ndgen, Li and Thomassen (2019) that every 4-chromatic planar graph has fractional chromatic number strictly greater than 3.",

keywords = "Arrangement of pseudolines and pseudocircles, Chromatic number, Critical graph, Fractional coloring, Triangle-saturated",

author = "Chiu, {Man Kwun} and Stefan Felsner and Manfred Scheucher and Felix Schr{\"o}der and Raphael Steiner and Birgit Vogtenhuber",

note = "Funding Information: Keywords: Arrangement of pseudolines and pseudocircles · Triangle-saturated · Chromatic number · Fractional coloring · Critical graph M.-K. Chiu was supported by ERC StG 757609. S. Felsner and M. Scheucher were supported by DFG Grant FE 340/12-1. M. Scheucher was supported by the internal research funding “Post-Doc-Funding” from Technische Universit{\"a}t Berlin. R. Steiner was supported by DFG-GRK 2434. B. Vogtenhuber was supported by the FWF project I 3340-N35. This work was initiated at a workshop of the collaborative DACH project Arrangements and Drawings in Malchow, Mecklenburg-Vorpommern. We thank the organizers and all the participants for the inspiring atmosphere. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

doi = "10.1007/978-3-030-83823-2_14",

language = "English",

series = "Trends in Mathematics",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "84--91",

booktitle = "Trends in Mathematics",

address = "Germany",

}