TY - JOUR

T1 - ON THE STRUCTURE OF GRAPHS WHICH ARE LOCALLY INDISTINGUISHABLE FROM A LATTICE

AU - Benjamini, Itai

AU - Ellis, David

PY - 2016/12

Y1 - 2016/12

N2 - For each integer d >= 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 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 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 Marquez 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.

AB - For each integer d >= 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 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 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 Marquez 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.

U2 - https://doi.org/10.1017/fms.2016.30

DO - https://doi.org/10.1017/fms.2016.30

M3 - مقالة

SN - 2050-5094

VL - 4

JO - FORUM OF MATHEMATICS SIGMA

JF - FORUM OF MATHEMATICS SIGMA

ER -