A note on upper embeddable graphs

AbstractWe define the maximum genus, γM(G), of a connected graph G to be the largest genus γ(N) for compact orientable 2-manifolds N in which G has a 2-cell imbedding. Several general results are established concerning the parameter γM(G), and the maximum genus of the complete graph Kn with n vertices is determined: γM(Kn) = (n − 1)(n − 2)

On 2-cell embeddings of graphs with minimum numbers of regions

AbstractLet G be a finite connected graph. The genus of G, denoted by γ(G), is the least integer n such that G can be imbedded in Sn. The maximum genus of G, denoted by γM(G), is the largest integer k such that G can be 2-cell imbedded in Sk. This paper characterizes those graphs G for which γ(G) = γM(G). As part of this characterization, it is shown that γM(G) = 0 if and only if G does not contain a subgraph isomorphic to a subdivision of one of two given graphs