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

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