Thin Hamiltonian cycles on Archimedean graphs

An Archimedean graph in R 2 is any finite subgraph of the vertices and edges of one of the 11 Archimedean tilings of R E shown in Figures 1-3. Archimedean filings are commonly characterized (as in Figures 1-3) by the order in which regular polygons (with unit edges) occur around each vertex of the tiling; see Grtinbaum and Shephard [3]. We use the same description for Archimedean subgraphs. When the Archimedean graph is a m × n rectangle from the 44 tiling, Fisher, Collins and Krompart [2] have noted that the area enclosed by a Hamiltonian cycle is a constant which depends on m and n, but not on the choice of the cycle. (This observation arose in enumerating Hamiltonian cycles on such m × n rectangles; see [5, 6] for related discussion and references.) We extend this to a more general class of Hamiltonian cycles on all Archimedean graphs. We say that a Hamiltonian cycle on an Archimedean graph is thin if the interior of the cycle contains no vertex of the underlying Archimedean tiling. An Archimedean graph (as in Figure 4) might contain no Hamiltonian cycle, or no thin Hamiltonian cycle (as in Figure 5), or a thin Hamiltonian cycle and one that is not thin (as in Figure 6). The following corollary of Pick's theorem shows that the area enclosed by any thin Hamilton cycle of a Archimedean graph of type 44 is a graph invariant (compare with [2]).