Inclusions and non-inclusions of Archimedean and Laves lattices

For each pair of graphs from among the Archimedean lattices and their dual Laves lattices we demonstrate that one is a subgraph of the other or prove that neither can be a subgraph of the other. Therefore, we determine the entire partial ordering by inclusion of these 19 infinite periodic graphs. There are a total of 72 inclusion relationships, of which 35 are covering relations in the partial ordering. Several elementary criteria are developed to prove non-inclusion results. The investigation is motivated by applications to the ordering of critical probabilities in bond and site percolation models, time constants in first-passage percolation theory, and connective constants in self-avoiding walks. the electronic journal of combinatorics 10 (2003), #R00 1