New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes

Abstract

In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph $c{r}_k(G)$ is the number of crossings required when every edge of $G$ must be drawn in one of $k$ distinct planes. It was shown in Czabarka et al. that $c{r}_2(Q_8)\le 256$ which we improve to $c{r}_2(Q_8)\le 128$. Our approach highlights the relationship between symmetric drawings and the study of $k$-planar crossing numbers. We conclude with several open questions concerning this relationship.