A result on combinatorial curvature for embedded graphs on a surface (Q998402)

Let \(G\) 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 \(x\) of \(G\) the combinatorial curvature is defined as \[ K_G(x)=1-\frac{d(x)}2+\sum\limits_{\sigma\in F(x)}\frac1{|\sigma|}\,, \] where \(d(x)\) is the degree of \(x,F(x)\) is the multiset of all open faces \(\sigma\) in the embedding such that the closure \(\bar{\sigma}\) contains \(x\), and \(|\sigma|\) is the number of sides of edges bounding the face \(\sigma\). In this paper, for a finite simple graph \(G\) embedded in a surface with \(3\leq d_G(x)<\infty\) and \(K_G(x)>0\) for all \(x\in V(G)\), it is proved that {\parindent=7mm \begin{itemize}\item[(i)]if \(G\) is embedded in a projective plane and \(|V(G)|=n\geq 290\), then \(G\) is isomorphic to \(P_n\); \item[(ii)]if \(G\) is embedded in a sphere and \(|V(G)|=n\geq 580\), then \(G\) is isomorphic to either the graph of an \(n\)-sided prism a \(A_n\) or an \(n\)-sided antiprisma \(B_n\). \end{itemize}}