Counting k-cycles in 5-connected planar triangulations

Key Points

• Every $5$-connected planar triangulation has at most $9n - 50$ cycles of length $5$ for $n \ge 20$.
• A constant $C(k)$ exists such that every $5$-connected planar graph has at most $C(k) \cdot n^{\lfloor{k/3}\rfloor}$ cycles of length $k \ge 6$.
• The upper bounds provided are asymptotically tight for cycles of lengths $5$ and $k \ge 6$.
• This work addresses the cycle counting problem in planar graphs, contributing to graph theory insights.