Triangles in Arrangements of Pseudocircles∗

Grünbaum conjectured that the number of triangular cells p3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n − 4. We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with p3(A)/n → 16/11 = 1.45. We conjecture that the lower bound p3 ≥ 4n/3 of Hershberger and Snoeyink is tight for infinitely many arrangements. For intersecting arrangements with digons we have p3 ≥ 2n/3, and conjecture that p3 ≥ n− 1. First counterexamples to Grünbaum’s conjecture were found on the basis of an exhaustive enumeration of all arrangements of n intersecting pseudocircles for n ≤ 7. It turned out that there is a unique digon-free intersecting arrangement N6 with n = 6 and only 8 triangles. This arrangement is a subarrangement of all minimizing examples for n = 7, 8, 9. We show that N6 is not circularizable, i.e., there is no equivalent arrangement of circles. These results suggest that Grünbaum’s conjecture might be true for digon-free intersecting arrangements of circles.