Gauss-Bonnet formula, finiteness condition, and characterizations for graphs embedded in surfaces

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 KG(x) = 1− d(x) 2 + ∑ σ∈F (x) 1 |σ| as that of [9], 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 σ̄ 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) |KG(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 ≤ dG(x) < ∞ and KG(x) > 0 for all x ∈ V (G), we have (i) if G is embedded in a projective plane and #(V (G)) = n ≥ 1722, then G is isomorphic to Pn; (ii) if G is embedded in a sphere and # ( V (G) ) = n ≥ 3444, then G is isomorphic to either An or Bn; and (iii) if dG(x) = 5 for all x ∈ V (G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs.