Counting $k$-cycles in $5$-connected planar triangulations

TL;DR

This paper establishes tight upper bounds on the number of cycles of specific lengths in 5-connected planar triangulations, revealing precise cycle count limits and their asymptotic behavior for large graphs.

Contribution

It provides the first tight bounds for the maximum number of 5-cycles and generalizes bounds for cycles of length k in 5-connected planar graphs.

Findings

Maximum of 9n-50 cycles of length 5 for n-vertex 5-connected planar triangulations.

Existence of constants C(k) bounding cycles of length k for large n.

Asymptotic tightness of bounds for all k ≥ 6.

Abstract

We show that every $n$-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all $n\ge 20$ and this upper bound is tight. We also show that for every $k\geq 6$, there exists some constant $C(k)$ such that for sufficiently large $n$, every $n$-vertex $5$-connected planar graph has at most $C(k) \cdot n^{\lfloor{k/3}\rfloor}$ many cycles of length $k$. This upper bound is asymptotically tight for all $k\geq 6$.