How to determine the maximum genus of a graph

Abstract

The maximum genus, γ M ( G), of a connected graph G is the largest genus γ( S) for orientable surfaces S in which G has a 2-cell embedding. In this paper, we define a new combinatorial invariant ξ( G), the Betti deficiency of G, to be ξ( C) = min C⊂ G { ξ( C) ‖ ξ( C) = number of odd components of a cotree C of G (by odd component we mean one with an odd number of edges). We formalize a new embedding technique to obtain the formula: γ M(G)= 1 2 (β(G)−ξ(G)) where β( G) denotes the Betti number of G. In a further paper, various consequences will be given.