The genus of the 2‐amalgamations of graphs

Abstract

Abstract A graph G is called the 2‐ amalgamation of subgraphs G 1 and G 2 if G = G 1 ∪ G 2 and G 1 ∩ G 2 = {x, y}, 2 distinct points. in this case we write G = G 1 ∪ {x, y} G 2 . in this paper we show that the orientable genus, γ( G ), satisfies the inequalities γ( G 1 ) + γ( G 2 ) − 1 ≤ γ( G 1 ∪ {x, y} G 2 ) ≤ γ( G 1 ) + γ( G 2 ) + 1 and that this is the best possible result, i. e., the resulting three values for γ( G 1 ∪ {x, y} G 2 ) which are possible can actually be realized by appropriate choices for G 1 and G 2 .