Genus bounds for embeddings with large minimum degree and representativity

Abstract

Let G be a simple graph, Σ be a surface, and Ψ:G→Σ be an embedding of G in Σ. The representativity ρ(Ψ) of the embedding is defined by ρ(Ψ)= min Γ{|Ψ(G)∩Γ|,Γ is a noncontractible simple closed curve in Σ}. An embedding with large representativity has a large locally planar region around each vertex. We provide a structure theorem for embeddings with large minimum degree and representativity and a lower bound for the genus of these embeddings in terms of the minimum degree and the representativity. Viewed slightly differently, this bound can be used to show that the representativity of graphs with minimum degree at least 7 embedded in the surface cannot be very large. More specifically, this representativity is essentially bounded above by the logarithm of the genus of the surface.