Arrangements of Pseudocircles: On Circularizability

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that every pair is either disjoint or intersects in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. Kang and Müller showed that every arrangement of at most 4 pseudocircles is circularizable. Linhart and Ortner found an arrangement of 5 pseudocircles which is not circularizable. In this paper we show that there are exactly four non-circularizable arrangements of 5 pseudocircles, exactly one of them is intersecting. For n = 6, we show that there are exactly three non-circularizable digon-free intersecting arrangements. We also have some additional examples of non-circularizable arrangements of 6 pseudocircles. The proofs of non-circularizability use various techniques, most depend on incidence theorems, others use arguments involving metric properties of arrangements of planes, or angles in planar figures. The claims that we have all non-circularizable arrangements with the given properties are based on a program that generated all connected arrangements of n ≤ 6 pseudocircles and all intersecting arrangements of n ≤ 7 pseudocircles. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed had to be examined by hand. One of the digon-free intersecting arrangements of pseudocircles with n = 6 is particularly interesting. It only has 8 triangles and occurs as a subarrangement of every digon-free intersecting arrangement with less than 2n− 4 triangles for n = 7, 8, 9. Hence it may be true that every digon-free intersecting arrangement of circles contains at least 2n− 4 triangles.