Certified Euclidean-Residue Minimal-Alignment Switch Decompositions for Three Edge-Disjoint Hamiltonian Cycles in Eisenstein--Jacobi Networks

Computer Science > Distributed, Parallel, and Cluster Computing [Submitted on 18 Jun 2026] Title:Certified Euclidean-Residue Minimal-Alignment Switch Decompositions for Three Edge-Disjoint Hamiltonian Cycles in Eisenstein--Jacobi Networks View PDF HTML (experimental)Abstract:Eisenstein--Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks. For a generator $\alpha=a+b\rho$, let $N=a^2+ab+b^2$ and $d=\gcd(a,b)$. If $d=1$, the three natural unit directions already give three edge-disjoint Hamiltonian cycles. If $d>1$, each unit direction splits into $d$ cycles and the EDHC problem becomes a cycle-splicing problem. Existing non-coprime EJ decompositions prove existence by using a rectangular representation and exchange schedules. This paper develops a different, local switch calculus in the natural Cayley geometry. The first two Hamiltonian cycles are built using the minimum possible $d-1$ intercomponent switches each, and the third factor is obtained as the unused edge complement. The contribution is deliberately not a new existence theorem for all non-coprime EJ networks; rather, it is a compact, formula-driven, minimal-switch decomposition for Euclidean-residue families whose complement incidence is proved symbolically. The proof separates four ingredients: component-label collapse, anchor cancellation, noncollision of lifted switch representatives, and connected complement incidence. No infinite-family theorem in this manuscript is proved by finite witnesses or by computational enumeration. The theorem scope is stated for the parameter ranges where an algebraic complement-incidence certificate is written down. Tables and CSV data are used only to verify and reproduce the formulas, never as proof of an infinite-family theorem. Submission history From: Bader Albader Dr. [view email][v1] Thu, 18 Jun 2026 06:25:42 UTC (7,202 KB) Current browse context: cs.DC References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.