Hypercube drawings with no long plane paths

TL;DR

This paper investigates the existence and limitations of plane substructures in hypercube graph drawings, constructing examples with no long plane paths or matchings and proving properties of rectilinear drawings in convex position.

Contribution

It introduces constructions of hypercube drawings lacking large plane subgraphs, and characterizes the structure of common plane subgraphs in all drawings, advancing understanding of hypercube graph embeddings.

Findings

Constructed hypercube drawings with no long plane paths or matchings.

Proved that convex-position rectilinear hypercube drawings contain long plane paths.

Showed that common plane subgraphs in large hypercube drawings are forests of caterpillars.

Abstract

We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges, and no plane matching of size more than $2d-4$. On the other hand, we prove that every rectilinear drawing of $Q_d$ with vertices in convex position contains a plane path of length $d$ (if $d$ is odd) or $d-1$ (if $d$ is even). We also prove that if a graph $G$ is a plane subgraph of every drawing of $Q_d$ for a sufficiently large $d$, then $G$ is necessarily a forest of caterpillars. Lastly, we give a short proof of a generalization of a result by Alpert et al. [Cong. Numerantium, 2009] on the maximum rectilinear crossing number of $Q_d$.