On the Existence of Partition of the Hypercube Graph into 3 Initial Segments
Ethan Soloway, Megan Triplett, Wenshi Zhao
TL;DR
This work addresses the problem of partitioning the hypercube $Q_n$ into three initial segments of prescribed sizes by introducing the notion of fit pairs $(a,b)$ for which there exist automorphisms $g_1,g_2\in Aut(Q_n)$ with $g_1(I_a) \cup g_2(I_b) = I_{a+b}$. It develops a novel, computable criterion based on the $m$-vector and $m$-table that reduces the fit condition to a vector equality under automorphisms, and proves the criterion via induction on $n$ using hyperface decompositions. Leveraging this main theorem, the authors classify all unfit pairs by exploiting translation and reflection symmetries, introducing a New Large Triangle of unfit pairs, and completing the remaining cases with detailed combinatorial analysis. They obtain a closed-form count $t_n = 4^n - \binom{4}{3}3^n + \binom{4}{2}2^n - \binom{4}{1}$ for the number of unfit pairs with $0<a,b\le 2^n$, and relate this to surjections from an $n$-element set onto a 4-element set, showing the density of unfit pairs approaches 1 as $n$ grows. The results illuminate the feasibility of 3-partitions of hypercubes and connect to edge isoperimetric-type questions in high-dimensional discrete geometry.
Abstract
Let $Q_n = \{0, 1\}^n$ be a hypercube graph. The initial segment $I_k \subseteq Q_n$ is the subset consisting of the first $k$ vertices of $Q_n$ in the binary order. A pair of integers $(a, b) \in \mathbb{Z}_{>0}^2$ is said to be fit if, whenever $2^n \geq a+b$, there exists $g_1, g_2 \in \text{Aut}(Q_n)$ such that $g_1(I_a) \cup g_2(I_b) = I_{a+b}$, and $(a,b)$ is unfit otherwise. For $a + b + c = 2^n$, there is a partition of $Q_n$ into $3$ initial segments of length $a, b$, and $c$ if and only if $(a, b)$ is a fit pair. Thus, the notion of fit and unfit pairs is closely related to the graph-partition problem for hypercube graphs. This paper introduces a new criterion in determining whether $(a,b)$ is fit using an easy-to-compute point-counting function and applies this criterion to generate the set of all unfit pairs. It further shows that the number of unfit pairs $(a,b)$, where $0 < a,b \leq 2^n$, is $4^n - \binom{4}{1}3^n + \binom{4}{2} 2^n - \binom{4}{1}$, which is also the number of surjection of an $n$-element set to a $4$-element set.