On the existence of Hamiltonian cycles in hypercubes

TL;DR

This paper characterizes when Hamiltonian cycles with fixed Euclidean distances exist in hypercubes, showing they occur precisely when the distance squared is an odd integer between 1 and k-1.

Contribution

It provides a complete characterization of Hamiltonian cycles in hypercubes with fixed Euclidean distances, extending to Euclidean leaper tours on hypercube graphs.

Findings

Hamiltonian cycles exist if and only if h is odd and 1 ≤ h ≤ k-1

Characterization of Euclidean leaper tours on hypercube graphs

Necessary and sufficient conditions for specific Euclidean distances in hypercube cycles

Abstract

Building on the results of our previous work on Euclidean leaper tours, considering all integers $k>1$ and $h>0$, we study the existence of Hamiltonian cycles in the vertex set $C(2,k):=\{0,1\}^k$ of the $k$-dimensional hypercube when the Euclidean distance between consecutive vertices is fixed. Since the distance between two vertices of $C(2,k)$ is $\sqrt{h}$ for some integer $h$, the problem amounts to determining for which integers $k$ and $h$ there exists a Hamiltonian cycle whose associated Euclidean distance is $\sqrt{h}$. In this paper, we prove that such cycles exist if and only if $h$ is odd and $1 \leq h \leq k-1$. As a result, for all integers $a \geq 0$, $b \geq a$ with $b>0$, we provide a necessary and sufficient condition for the existence of closed Euclidean $(a,b)$-leaper tours on $2 \times 2 \times \cdots \times 2$ chessboards, where the associated distance equals $\sqrt{a^2+b^2}$.