A result on combinatorial curvature for embedded graphs on a surface

Let GG be an infinite graph embedded in a surface 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 xx of GG, we define the combinatorial curvature KG(x)=1−d(x)2+∑σ∈F(x)1|σ|, where d(x)d(x) is the degree of xx, F(x)F(x) is the multiset of all open faces σσ in the embedding such that the closure σ̄ contains xx, and |σ||σ| is the number of sides of edges bounding the face σσ. In this paper, for a finite simple graph GG embedded in a surface with 3≤dG(x)<∞3≤dG(x)<∞ and KG(x)>0KG(x)>0 for all x∈V(G)x∈V(G), we have (i) if GG is embedded in a projective plane and |V(G)|=n≥290|V(G)|=n≥290, then GG is isomorphic to PnPn; (ii) if GG is embedded in a sphere and |V(G)|=n≥580|V(G)|=n≥580, then GG is isomorphic to either AnAn or BnBn.