Parity balance of the $i$-th dimension edges in Hamiltonian cycles of the hypercube

TL;DR

This paper investigates the parity distribution of dimension-specific edges in Hamiltonian cycles of hypercubes, proving a balance property, and explores a conjecture about inscribed squares, confirming it for small dimensions and certain conditions.

Contribution

It introduces and proves a parity balance property for edges in Hamiltonian cycles of hypercubes and explores the inscribed squares conjecture, providing partial results and new concepts like the equi-independence number.

Findings

Parity of i-th dimension edges is balanced in Hamiltonian cycles.

Confirmed the inscribed squares conjecture for n ≤ 7.

Proved the conjecture for cycles with many edges in the same dimension.

Abstract

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge $\edg{v_1}{v_2}$ is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs.