The orientable genus is nonadditive
Dan Archdeacon · Journal of Graph Theory · 1986
Abstract A graph G is a k‐amalgamation of two graphs G1 and G2 if G = G1 ∪ G2 and G1 ∩ G2 is a set of k vertices. In this paper we construct 3‐amalgamations Gn = Hn ∪ Hn such that γ(Gn) = 5n and γ(Hn) = 3n, where γ denotes the orientable genus of a graph. Thus γ(G1 ∪ G2) may differ from γ(G1) + γ(G2) by an arbitrarily large amount for amalgamations over 3 (or more) vertices. In contrast, an earlier paper shows that the nonorientable genus of a k‐amalgamation differs from the sum of the nonorientable genera of its parts by at most a quadratic on k.