On the structure of graphs which are locally indistinguishable from a lattice

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional square lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d \geq 3$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They may be viewed as quotient lattices of $\mathbb{L}^d$ in various compact orbifolds.) In the $d=2$ case, our methods yield new proofs of structure theorems of Thomassen and of M\'arquez, de Mier, Noy and Revuelta, and also yield short, `algebraic' restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory.