TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 7, July 2007, Pages 3287–3300 S 0002-9947(07)04125-6 Article electronically published on February 21, 2007 AN ANALOGUE OF THE DESCARTES-EULER FORMULA FOR INFINITE GRAPHS AND HIGUCHI’S CONJECTURE MATT DEVOS AND BOJAN MOHAR Abstract. 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 R 3 . 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. 1. Introduction A polygonal surface R is a simply connected 2-manifold without boundary which is obtained from a (possibly infinite) collection of disjoint simple polygons in R 2 by identifying them along edges of equal length. Based on this construction, R may be viewed as an embedded graph, and accordingly, we equip it with three distinguished sets: vertices , edges , and faces , respectively denoted V (R), E(R), and F (R), and defined to be the set of all subsets of R which correspond (respectively) to a vertex of a polygon, edge of a polygon, or polygon itself. We view vertices, edges, and faces both combinatorially and as subsets of R; two such objects are defined to be incident if one is a proper subset of the other. The space R is also equipped with a natural intrinsic metric: the distance between two points is the length of the shortest rectifiable curve joining them. This metric gives each point x ∈R a Gaussian curvature which we denote by κ(x). If x is not a vertex, then κ(x) = 0. If x is a vertex, then κ(x) is equal to 2π minus the sum over all faces f incident Received by the editors July 2, 2004 and, in revised form, May 11, 2005. 2000 Mathematics Subject Classification. Primary 05C10. The first author was supported in part by the SLO-USA Grant BI-US/04-05/36 and by the Slovenian grant L1–5014. The second author was supported in part by the Ministry of Education, Science and Sport of Slovenia, Research Program P1–0297 and Research Project J1–6150. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 3287 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use