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

TL;DR

This paper surveys recent developments on inequalities relating skewness, crossing number, and Euler's bound for graphs on surfaces, and presents new results for the folded cube's genus.

Contribution

It introduces new results for the folded cube's genus and discusses the inequalities involving skewness, crossing number, and Euler's bound for graphs on surfaces.

Findings

New bounds for the folded cube's genus.

Survey of recent inequalities for graphs on surfaces.

Extended understanding of skewness and crossing number relations.

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.