How Low Can You Go? 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 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 [2] 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