Document Type
Lecture
Publication Date
4-14-2004
Abstract
Let G be a plane graph (finite or infinite) such that (1) G is locally finite and (2) every face of G is bounded by a cycle. Then the combinatorial curvature of G is the function Φ(G) : V(G) −→ R such that for any x ∈ V(G), Φ(x) = 1−d(x)/2+P x∈F 1/|F|, where the summation is taken over all facial cycles of G containing x. The curvature interprets the degree of difficulty of tiling the plane at x and it is dual of another curvature introduced by Gromov. Higuchi proved that there is a negative real number µ such that Φ(x) < µ if Φ(x) < 0 and the positive curvature can be arbitrarily small. We show that if Φ(x) ≥ 0 then P x∈V(G) Φ(x) is bounded if and only if there are only finite number x such that Φ(x)= 0. Higuchi also conjectured that G is f inite if Φ(x) > 0 for all x. Sun and Yu proved this for cubic plane graphs. We completely characterized finite graphs with positive curvatures provide the number of vertices is large.
Relational Format
presentation
Recommended Citation
Chen, Guantao, "Plane Graphs with Positive Curvature" (2004). Combinatorics Seminar. 99.
https://egrove.olemiss.edu/math_combinatorics/99
