ON THE STRUCTURE OF GRAPHS WHICH ARE LOCALLY INDISTINGUISHABLE FROM A LATTICE

Abstract

For each integer $d\geqslant 3$, we obtain a characterization of all graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius 3 in $\mathbb{L}^{d}$, the graph of the $d$-dimensional integer lattice. The finite, connected graphs with this property have a highly rigid, ‘global’ algebraic structure; they can be viewed as quotient lattices of $\mathbb{L}^{d}$ in various compact $d$-dimensional orbifolds which arise from crystallographic groups. We give examples showing that ‘radius 3’ cannot be replaced by ‘radius 2’, and that ‘orbifold’ cannot be replaced by ‘manifold’. In the $d=2$ case, our methods yield new proofs of structure theorems of Thomassen [‘Tilings of the Torus and Klein bottle and vertex-transitive graphs on a fixed surface’, Trans. Amer. Math. Soc.323 (1991), 605–635] and of Márquez et al. [‘Locally grid graphs: classification and Tutte uniqueness’, Discrete Math.266 (2003), 327–352], and also yield short, ‘algebraic’ restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, geometry and group theory.