New results and problems on crossing numbers

Abstract

The purpose of this presentation is threefold. We start by giving a set of new necessary and sufficient conditions for the Zarankiewicz conjecture on the crossing number of complete bipartite graphs. These conditions are expressed in terms of the crossing numbers' divisibility properties and their expressibility as polynomials. We then examine the Cartesian Product of two graphs, and give an alternative proof to that given by Klesc regarding the crossing number of a particular family of graphs. In the last section, we consider Generalized Petersen Graphs and calculate for the first time the crossing number of another family of graphs.