Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Abstract

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature

$$\Phi_G(x) = 1 - \frac{d(x)}{2} + \sum_{\sigma \in F(x)} \frac{1}{|\sigma|}$$

as that of [8], where d(x) is the degree of x, F(x) is the multiset of all open faces σ in the embedding such that the closure $\bar\sigma$ contains x (the multiplicity of σ is the number of times that x is visited along ∂σ), and |σ| is the number of sides of edges bounding the face σ. In this paper, we first show that if the absolute total curvature ∑ _x∈V(G) |Φ _G (x)| is finite, then G has only finite number of vertices of non-vanishing curvature. Next we present a Gauss-Bonnet formula for embedded infinite graphs with finite number of accumulation points. At last, for a finite simple graph G with 3 ≤ d _G (x) < ∞ and Φ _G (x) > 0 for every x ∈ V(G), we have (i) if G is embedded in a projective plane and #(V(G)) = n ≥ 1722, then G is isomorphic to the projective wheel P _n ; (ii) if G is embedded in a sphere and #(V(G)) = n ≥ 3444, then G is isomorphic to the sphere annulus either A _n or B _n ; and (iii) if d _G (x) = 5 for all x ∈ V(G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs.