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.
Submitted by: Dan Archdeacon (with thanks to Yusuke Higuchi email@example.com)
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