Hypercube drawings with no long plane paths

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 $Q_d$.

Figures (7)

• Figure 1: The length of $e$ is $7$: the number of convex hull edges within the depicted arc.
• Figure 2: The construction of $\mathcal{H}_4$ (right) from $\mathcal{H}_3$ (left). In the right picture, the edges in $\mathcal{H}'$ and $\mathcal{H}"$ are depicted in black and green, respectively, and the red edges depict the ones between these two copies of $\mathcal{H}_3$.
• Figure 3: A plane path of length $7$ in $\mathcal{H}_5$ is highlighted.
• Figure 4: The alternative construction of $\mathcal{H}_d$.
• Figure 6: A plane matching of length $6$ in $\mathcal{H}_5$ from the proof of Proposition \ref{['prop:existencematching']} is highlighted in grey and vertices $v_0,\dots,v_3$ and $w_0,\dots,w_3$ are marked. The two edges highlighted in cyan form a parallel pair that is uncrossed by the matching and can be added to form a larger plane subgraph.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 6
• Proposition 6
• Lemma 6: lemma_longplanepaths
• Theorem 6