**Combinatorial
Curvature**

Let G be a (possibly infinite) 2-connected graph embedded on the
plane. Suppose that a) each
vertex v is of finite degree deg(v) at
least 3, and b) each face f is incident with a finite number deg(f) > 3 of
edges. Define the combinatorial curvature as the function Psi from vertices to integers as Psi(v) = 1 - (deg(v)/2) + Sum (1/deg(f))
where the sum is taken over all faces f
incident with v. For example,
in the dodecahedron, the combinatorial curvature at each vertex x is 1 - 3/2 + 3(1/5) = 1/10.

The concept was originally due to Ishida [I]. In [H] Yusuke Higuchi
showed that for any infinite planar graph G there exists a positive number epsilon (depending on G), such that if Psi(v) < 0, then Psi (v) < - epsilon.

**Conjecture:*** If *Sum(Psi(v)) = 2* (where the
sum is over all vertices) and *Psi(v) > 0* for all vertices,
then *G* is finite. *

References:

[H] Y. Higuchi, Combinatorial curvature
for planar graphs, J. of Graph Theory
38 (2001) 220-229.

[I] M. Ishida, Pseudo-curvature of a graph, in a lecture at the `Workshop on topological graph theory', Yokohama National University, 1990.

[I] M. Ishida, Pseudo-curvature of a graph, in a lecture at the `Workshop on topological graph theory', Yokohama National University, 1990.

Submitted by: Dan Archdeacon (with thanks to Yusuke Higuchi *higuchi@cas.showa-u.ac.jp*)

Send comments to *dan.archdeacon@uvm.edu*

November, 2003