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.