On the existence of Hamiltonian cycles in hypercubes
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 √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 √h. In this paper, we prove that such cycles exist if and only if h is odd and 1 ≤ h ≤ k-1. As a result, for all integers a ≥ 0, b ≥ a with b > 0, we provide a necessary and sufficient condition for the existence of closed Euclidean (a,b)-leaper tours on 2 × 2 × ⋯ × 2 chessboards, where the associated distance equals √{a^2+b^2}.
Fields of Study & Subjects
Interconnection Networks and Systems
Optimization and Search Problems
Advanced Graph Theory Research
Publication details
- Journal / source
- Notes on Number Theory and Discrete Mathematics
- Publisher
- Prof. Marin Drinov Publishing House of BAS (Bulgarian Academy of Sciences)
- Volume
- 32
- Issue
- 1
- Pages
- 198–206
- Publication date
- 18 March 2026
- Language
- EN