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

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.