The orientable genus is nonadditive

A graph G is a k ‐amalgamation of two graphs G 1 and G 2 if G = G 1 ∪ G 2 and G 1 ∩ G 2 is a set of k vertices. In this paper we construct 3‐amalgamations G n = H n ∪ H n such that γ( G n ) = 5 n and γ( H n ) = 3 n , where γ denotes the orientable genus of a graph. Thus γ( G 1 ∪ G 2 ) may differ from γ( G 1 ) + γ( G 2 ) 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 .