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

TL;DR

This paper improves the upper bound on the biplanar crossing number of the 8-dimensional hypercube from 256 to 128, using symmetric drawings and exploring the relationship between graph symmetry and crossing numbers.

Contribution

It presents a tighter upper bound on the biplanar crossing number of Q8 and emphasizes the connection between symmetric graph drawings and k-planar crossing numbers.

Findings

Improved upper bound on cr_2(Q_8) from 256 to 128

Highlights the link between symmetric drawings and crossing number studies

Raises open questions about the relationship between symmetry and crossing numbers

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 Czabarka et al. that $cr_2(Q_8) \leq 256$ which we improve to $cr_2(Q_8) \leq 128$. 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.