# Highly-connected planar cubic graphs with few or many Hamilton cycles

## TL;DR

This paper investigates the number of Hamilton cycles in highly-connected planar cubic graphs, introducing new extremal families, developing counting methods inspired by physics, and exploring graphs with both many and few Hamilton cycles.

## Contribution

It constructs new extremal families of planar cubic graphs with many or few Hamilton cycles and develops transfer matrix inspired methods for counting Hamilton cycles in nanotubes.

## Key findings

- Families of graphs with more Hamilton cycles than Petersen graphs.
- An infinite family of graphs with exactly 4 Hamilton cycles.
- Partial results and conjectures on extremal Hamilton cycle counts.

## Abstract

In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest and almost longest cycles in cubic graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these families. In particular, we find families of cyclically $5$-edge connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes --- more precisely, the family consists of the zigzag nanotubes of (fixed) width $5$ and increasing length. In order to count the Hamilton cycles in the nanotubes, we develop methods inspired by the transfer matrices of statistical physics. We outline how these methods can be adapted to count the Hamilton cycles in nanotubes of greater (but still fixed) width, with the caveat that the resulting expressions involve matrix powers. We also consider cyclically $4$-edge-connected cubic planar graphs with few Hamilton cycles, and exhibit an infinite family of such graphs each with exactly $4$ Hamilton cycles. Finally we consider the "other extreme" for these two classes of graphs, thus investigating cyclically $4$-edge connected cubic planar graphs with many Hamilton cycles and the cyclically $5$-edge connected cubic planar graphs with few Hamilton cycles. In each of these cases, we present partial results, examples and conjectures regarding the graphs with few or many Hamilton cycles.