Coloring circle arrangements: New 4-chromatic planar graphs

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of (pseudo-)circle arrangements. In this paper the conjecture is verified for △-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps △-saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This corona construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture. We further investigate fractional colorings. It is shown that the arrangement graph of every arrangement A of pairwise intersecting pseudocircles is “close” to being 3-colorable. More precisely, the fractional chromatic number χf(A) of the arrangement graph is bounded from above by χf(A)≤3+O(1n), where n is the number of pseudocircles of A. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019).