On the Existence of Partition of the Hypercube Graph into 3 Initial Segments

TL;DR

This paper investigates the conditions under which the hypercube graph can be partitioned into three initial segments, introducing a new criterion to identify unfit pairs and deriving a formula for their count.

Contribution

It presents a new, easy-to-compute criterion for determining fit pairs and characterizes all unfit pairs, linking the problem to set surjections.

Findings

Derived a formula for the number of unfit pairs as a polynomial in n.

Established a criterion based on point counting for fit pairs.

Connected the problem to the enumeration of surjections from an n-set to a 4-set.

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.