New Bounds on the Biplanar Crossing Number of Low-Dimensional Hypercubes: How Low Can You Go?

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 cr_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 cr_2(Q_8) ≤ 256 which we improve to cr_2(Q_8) ≤ 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.