New Bounds on the Biplanar and $k$-Planar Crossing Numbers

The biplanar crossing number of a graph $G$ is the minimum number of crossings over all possible drawings of the edges of $G$ in two disjoint planes. We present new bounds on the biplanar crossing number of complete graphs and complete bipartite graphs. In particular, we prove that the biplanar crossing number of complete bipartite graphs can be approximated to within a factor of $3$, improving over the best previously known approximation factor of $4.03$. For complete graphs, we provide a new approximation factor of $3.17$, improving over the best previous factor of $4.34$. We provide similar improved approximation factors for the $k$-planar crossing number of complete graphs and complete bipartite graphs, for any positive integer $k$. We also investigate the relation between (ordinary) crossing number and biplanar crossing number of general graphs in more depth, and prove that any graph with a crossing number of at most $10$ is biplanar.