An Analogue of the Descartes-euler Formula for Infinite Graphs and Higuchi’s Conjecture

Let R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V be the set of vertices, and for every v ∈ V , let κ(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that ∑ v∈V κ(v) = 4π whenever R may be realized as the surface of a convex polytope in R3. More generally, if R is made of finitely many polygons, Euler’s formula is equivalent to the equation ∑ v∈V κ(v) = 2πχ(R) where χ(R) is the Euler characteristic of R. Our main theorem shows that whenever ∑ v∈V :κ(v)<0 κ(v) converges and there is a positive lower bound on the distance between any pair of vertices in R, there exists a compact closed 2-manifold S and an integer t so that R is homeomorphic to S minus t points, and further ∑ v∈V κ(v) ≤ 2πχ(S)− 2πt. In the special case when every polygon is regular of side length one and κ(v) > 0 for every vertex v, we apply our main theorem to deduce that R is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless R is a prism, antiprism, or the projective planar analogue of one of these that |V | ≤ 3444. This resolves a recent conjecture of Higuchi.