Slicing all Edges of an $n$-cube Requires $n^{2/3}$ Hyperplanes

TL;DR

This paper establishes a lower bound on the number of hyperplanes needed to dissect all edges of an n-cube graph, improving previous bounds and advancing understanding of geometric graph partitioning.

Contribution

It proves a new lower bound of n^{2/3} hyperplanes, strengthening the theoretical understanding of hyperplane cuts in high-dimensional cubes.

Findings

At least n^{2/3} hyperplanes are necessary to dissect all edges.

The bound improves previous results from  n^{0.51} to n^{2/3}.

The result advances the theoretical limits of geometric graph partitioning.

Abstract

Consider the $n$-cube graph with vertices $\{-1,1\}^n$ and edges connecting vertices with hamming distance $1$. How many hyperplanes in $\mathbb{R}^n$ are needed in order to dissect all edges? We show that at least $\widetilde{\Omega}(n^{2/3})$ are needed, which improves the previous bound of $\Omega(n^{0.51})$ by Yehuda and Yehudayoff.