Counting cycles in planar triangulations

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is Omega(n) for any cycle length at most 3 + max{rad(G & lowast;), [(n-3 radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian n-vertex triangulations containing O(n) many k-cycles for any k E {[n - 5 root n1, ... , n}. Furthermore, we prove that planar 4connected n-vertex triangulations contain Omega(n) many k-cycles for every k E {3,. .., n}, and that, under certain additional conditions, they contain Omega(n2) k-cycles for many values of k, including n. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar