# Planar cubic graphs of small diameter

## TL;DR

This paper constructs a family of cubic planar graphs with faces of length at most 7 that have logarithmic diameter, challenging previous beliefs about diameter bounds in such graphs.

## Contribution

It introduces a new family of cubic planar graphs with small faces and logarithmic diameter, countering prior conjectures about diameter growth.

## Key findings

- Cubic planar graphs with faces of length at most 7 can have diameter in O(log n)
- Counterexample to the suspicion that such graphs have diameter in Omega(√n)
- Provides explicit construction of these graphs

## Abstract

Cubic planar $n$-vertex graphs with faces of length at most $6$, e.g., fullerene graphs, have diameter in $\Omega(\sqrt{n})$. It has been suspected, that a similar result can be shown for cubic planar graphs with faces of bounded length. This note provides a family of cubic planar $n$-vertex graphs with faces of length at most $7$ and diameter in ${O}(\log n)$, thus refuting the above suspicion.