Counting cycles in planar triangulations

Abstract

Listen

Translate article

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is Ω(n) for any cycle length at most 3+max⁡{rad(G⁎),⌈(n−32)log3⁡2⌉}, where rad(G⁎) denotes the 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∈{⌈n−n5⌉,…,n}. Furthermore, we prove that planar 4-connected n-vertex triangulations contain Ω(n) many k-cycles for every k∈{3,…,n}, and that, under certain additional conditions, they contain Ω(n2)k-cycles for many values of k, including n.