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

By Matt DeVos, Bojan Mohar. 2007

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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 k(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that Συ∈v k(υ) = 4π whenever R, may be realized as the surface of a convex polytope in R 3 . More generally, if R is made of finitely many polygons, Euler's formula is equivalent to the equation Συ∈vκ(υ) = 1πΧ(R) where Χ(R) is the Euler characteristic of R. Our main theorem shows that whenever Συ∈V:κ(υ) 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 Ρ is a prism, antiprism, or the projective planar analogue of one of these that |V| < 3444. This resolves a recent conjecture of Higuchi.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/03319b7bb84a95e786e97e884b4d1af517717c70

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