# Gaps in the cycle spectrum of 3-connected cubic planar graphs

## TL;DR

This paper proves that large 3-connected cubic planar graphs have cycles within a specific length range, and demonstrates near-optimal bounds by constructing graphs lacking cycles in a slightly smaller range.

## Contribution

It establishes a universal cycle length interval for large 3-connected cubic planar graphs and shows the bounds are nearly tight through explicit constructions.

## Key findings

- Large graphs have cycles in [k, 2k+9] for any k.
- Constructed graphs lack cycles in [k, 2k+1] for even k ≥ 4.
- Bounds are close to optimal, indicating limits of cycle lengths.

## Abstract

We prove that, for every natural number $k$, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in $[k,2k+9]$. We also show that this bound is close to being optimal by constructing, for every even $k\geq 4$, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in $[k,2k+1]$.