Maximum genus, connectivity and minimal degree of graphs

AbstractThis paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form:Let G be a k-edge-connected graph with minimum degree δ, for each positive integer k(⩽3), there exists a non-decreasing function f(δ) such that the maximum genus γM(G) of G satisfies the relation γM(G)⩾f(δ)β(G), and furthermore that limδ→∞f(δ)=1/2, where β(G)=|E(G)|-|V(G)|+1 is the cycle rank of G.The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors