Maximum genus, connectivity, and Nebeský's Theorem

Dan Archdeacon,Roman Nedela,Martin Škoviera,Michal Kotrbčík

doi:10.26493/1855-3974.356.66e

Abstract

We prove lower bounds on the maximum genus of a graph in terms of its connectivity and Betti number (cycle rank). These bounds are tight for all possible values of edge-connectivity and vertex-connectivity and for both simple and non-simple graphs. The use of Nebeský's characterization of maximum genus gives us both shorter proofs and a description of extremal graphs. An additional application of our method shows that the maximum genus is almost additive over the edge cuts.

Highlights

We study cellular embeddings of graphs in orientable surfaces of large genus
Suppose that G is a connected graph with v vertices and e edges embedded with f faces on an orientable surface of genus g, denoted here by Sg
The EulerPoincareformula asserts that v − e + f = 2 − 2g. By combining this formula with the formula β(G) = 1 − v + e for the Betti number of G we get f − 1 = β − 2g

Summary

Introduction

ΓM (G), which is the maximum value of g over all cellular embeddings of G. An embedding of a graph with odd Betti number always has at least two faces and deficiency at least one. In this paper we give lower bounds N on the maximum genus of graphs in various classes C defined by the graph’s Betti number and edge-connectivity. Nebesky’s Theorem asserts that there is a set A of edges in a maximum genus embedding that reverses the steps just described; the proof in this direction is more difficult. Previous results relating maximum genus, connectivity, and the Betti number mostly used Xuong’s Theorem [15] to determine the maximum genus It is our use of Nebesky’s alternative characterization [12] that gives us new insights and shorter proofs; these techniques were first introduced in an earlier unpublished version of this paper (1996)

Graphs with given connectivity

Edge-cuts