Maximum genus, connectivity, and Nebesky's Theorem

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 Nebesky'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.