# Least-perimeter partition of the disc into $N$ regions of two different areas

## TL;DR

This paper conjectures optimal partitions of a disk into N regions of two areas, using graph enumeration and numerical methods to identify least-perimeter configurations for N up to 10.

## Contribution

It introduces a method to identify candidate least-perimeter partitions for two-area regions in a disk by enumerating three-connected cubic graphs and interpolating perimeter data.

## Key findings

- Candidates are identified for N ≤ 10 regions.
- Optimal configurations depend on the area ratio.
- Candidates are best for specific area ratios at larger N.

## Abstract

We present conjectured candidates for the least perimeter partition of a disc into $N \le 10$ regions which take one of two possible areas. We assume that the optimal partition is connected, and therefore enumerate all three-connected simple cubic graphs for each $N$. Candidate structures are obtained by assigning different areas to the regions: for even $N$ there are $N/2$ regions of one area and $N/2$ regions of the other, and for odd $N$ we consider both cases, i.e. where the extra region takes either the larger or the smaller area. The perimeter of each candidate is found numerically for a few representative area ratios, and then the data is interpolated to give the conjectured least perimeter candidate for all possible area ratios. At larger $N$ we find that these candidates are best for a more limited range of the area ratio.