An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture

Abstract

Let R \mathcal {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 V be the set of vertices, and for every v ∈ V v \in V , let κ ( v ) \kappa (v) denote the (Gaussian) curvature of v v : 2 π 2 \pi minus the sum of incident polygon angles. Descartes showed that ∑ v ∈ V κ ( v ) = 4 π \sum _{v \in V} \kappa (v) = 4 \pi whenever R \mathcal {R} may be realized as the surface of a convex polytope in R 3 \mathbb {R}^3 . More generally, if R \mathcal {R} is made of finitely many polygons, Euler’s formula is equivalent to the equation ∑ v ∈ V κ ( v ) = 2 π χ ( R ) \sum _{v \in V} \kappa (v) = 2 \pi \chi (\mathcal {R}) where χ ( R ) \chi (\mathcal {R}) is the Euler characteristic of R \mathcal {R} . Our main theorem shows that whenever ∑ v ∈ V : κ ( v ) > 0 κ ( v ) \sum _{v \in V : \kappa (v) > 0} \kappa (v) converges and there is a positive lower bound on the distance between any pair of vertices in R \mathcal {R} , there exists a compact closed 2-manifold S \mathcal {S} and an integer t t so that R \mathcal {R} is homeomorphic to S \mathcal {S} minus t t points, and further ∑ v ∈ V κ ( v ) ≤ 2 π χ ( S ) − 2 π t \sum _{v \in V} \kappa (v) \le 2 \pi \chi (\mathcal {S}) - 2 \pi t . In the special case when every polygon is regular of side length one and κ ( v ) > 0 \kappa (v) > 0 for every vertex v v , we apply our main theorem to deduce that R \mathcal {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 \mathcal {R} is a prism, antiprism, or the projective planar analogue of one of these that | V | ≤ 3444 |V| \le 3444 . This resolves a recent conjecture of Higuchi.