Coloring circle arrangements: New $4$-chromatic planar graphs

TL;DR

This paper investigates the colorability of arrangement graphs of (pseudo-)circles, verifying a conjecture for certain classes, proposing strengthenings, and exploring fractional colorings, including counterexamples to previous conjectures.

Contribution

It proves the conjecture for specific arrangements, introduces a new construction linking arrangements with different chromatic properties, and disproves a previous conjecture on fractional colorability.

Findings

Verified the conjecture for $ riangle$-saturated arrangements.

Introduced a 'corona' construction linking arrangements with 4-chromatic graphs.

Disproved the conjecture that all 4-regular planar graphs are not fractionally 3-colorable.

Abstract

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 arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-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 $\triangle$-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 also investigate fractional colorings. It is shown that the arrangement graph of every arrangement $\mathcal{A}$ of pairwise intersecting pseudocircles is "close" to being $3$-colorable. More precisely, the fractional chromatic number $\chi_f(\mathcal{A})$ of the arrangement graph is bounded from above by $\chi_f(\mathcal{A}) \le 3+O(\frac{1}{n})$, where $n$ is the number of pseudocircles of $\mathcal{A}$. 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, K\"{u}ndgen, Li, and Thomassen (2019).