Skewness, crossing number and Euler's bound for graphs on surfaces

Abstract

For every connected graph $G$ and surface $S$, we consider the well-known string of inequalities $\delta_S(G) \leq \mu_S(G) \leq \nu_S(G)$, where $\mu$ and $\nu$ denote skewness and crossing number and $\delta$ is the Euler-formula lower bound. Recent developments are surveyed; new results are given for the ``folded'' cube including its genus.