Curvature and Geometry of Tessellating Plane Graphs

Abstract

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesics holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where ( p, q) ∈ {(3, 6), (4, 4), (6, 3)}.

Cheeger constants, growth and spectrum of locally tessellating planar graphs

“…Now we present the conditions which have to be satisfied that a planar graph is locally tessellating: Note that these properties force the graph G to be connected. Examples are tessellations R 2 introduced in [BP1,BP2], trees in R 2 , and particular finite tessellations on the sphere mapped to R 2 via stereographic projection.…”

Distance and routing labeling schemes for non-positively curved plane graphs

“…for example [4,23,24]. From the Gauss-Bonnet formula it follows that a plane graph of non-positive curvature has at least 3 corners.…”

“…A plane graph G has non-positive curvature if κ(v) 0 for every inner vertex v of G. It can be easily shown that the plane graphs of each of the types (4,4), (3,6), and (6, 3) have non-positive curvature, and from this perspective they have been investigated in a number of papers; cf. for example [4,23,24].…”

“…Notice that the class of (4, 4)-graphs is self-dual in the sense that the geometric dual of a (4, 4)-graph is again a (4, 4)-graph, while the classes of (3, 6)-and (6, 3)-graphs are mutually dual. Two neighbors x, y of a vertex v of G are called consecutive if v, x, y belong to a common inner face of G. Following [4,22,24] and the references therein, we introduce now the curvature function of a plane graph G. Assume that each inner face with k sides of G is viewed as a regular k-gon in Euclidean plane with side length 1. For a vertex v of G, let α(v) denote the sum of the corner angles of the regular polygons containing the vertex v. If v is an inner vertex of G, denote the curvature at v to be κ(v) = 2π − α(v), i.e., it is defined as the 2π -angle-defect of the flat polygons meeting at v. When v is a vertex in the boundary ∂G, define the turning angle at v to be τ (v) = π − α(v).…”

“…In this note we design distance and routing labeling schemes for three natural classes of planar graphs introduced in [23,24] and further investigated in [3,4,29,32] and the references cited therein. These are the basic classes of planar graphs of non-positive combinatorial curvature:…”