Extremal embedded graphs

Abstract

Let G be a ribbon graph and μ( G ) be the number of components of the virtual link formed from G as a cellularly embedded graph via the medial construction. In this paper we first prove that μ( G ) ≤ f ( G ) + γ( G ) , where f ( G ) and γ( G ) are the number of boundary components and Euler genus of G , respectively. A ribbon graph is said to be extremal if μ( G ) = f ( G ) + γ( G ) . We then obtain that a ribbon graph is extremal if and only if its Petrial is plane. We introduce a notion of extremal minor and provide an excluded extremal minor characterization for extremal ribbon graphs. We also point out that a related result in the monograph by Ellis-Monaghan and Moffatt is not correct and prove that two related conjectures raised by Huggett and Tawfik hold for more general ribbon graphs.