Counting cycles in planar triangulations

Abstract

We investigate the minimum number of cycles of specified lengths in planar n-vertex triangulations G. We prove that this number is $\Omega \left(n\right)$ for any cycle length at most $3+\max \left\{\mathrm{rad}\right(G^{⁎}),\lceil {\left(\frac{n-3}{2}\right)}^{{\log }_{3}2}\rceil \}$, where $\mathrm{rad}\left(G^{⁎}\right)$ 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 $O\left(n\right)$ many k-cycles for any $k\in \{\lceil n-\sqrt[5]{n}\rceil ,\dots ,n\}$. Furthermore, we prove that planar 4-connected n-vertex triangulations contain $\Omega \left(n\right)$ many k-cycles for every $k\in \{3,\dots ,n\}$, and that, under certain additional conditions, they contain $\Omega \left(n^2\right)$ k-cycles for many values of k, including n.