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Constructing Hamiltonian Decompositions of Complete $k$-Uniform Hypergraphs

Javad Maheri, Petros Elia

TL;DR

The paper addresses the problem of partitioning the edge set of the complete $k$-uniform hypergraph $K_n^k$ into Hamiltonian cycles, extending previous existence results to explicit designs for all $k$ with prime $n$ and $k< n/2$. It introduces a generator-based framework that maps each $k$-subset to a Hamiltonian cycle via a generator vector $oldsymbol{g}$, a representative vector $oldsymbol{r}_{oldsymbol{g}}$, and the cycle $oldsymbol{ ext{S}}_{oldsymbol{g}}$, with a cycle period equal to the number of vertices $n$ when $ abla$ conditions hold (e.g., $ ext{gcd}(n,k)=1$). By constructing a comprehensive set of generators $oldsymbol{G}_{ ext{tot}}$, determined from parameters $oldsymbol{ extsigma}_{ ext{min}}=k-1$ and $oldsymbol{ extsigma}_{ ext{max}}=n-ig\lceil n/kig ceil$, the authors ensure that the corresponding vector sets partition $oldsymbol{A}_{n,k}$ into $N=inom{n}{k}/n$ disjoint Hamiltonian cycles, providing an explicit Hamiltonian decomposition. The work delivers concrete constructions for all $k$ with prime $n$, with a detailed example for $(n,k)=(11,5)$ and discussion of practical implications for distributed systems, coded caching, routing, and fault tolerance.

Abstract

Motivated by the wide-ranging applications of Hamiltonian decompositions in distributed computing, coded caching, routing, resource allocation, load balancing, and fault tolerance, our work presents a comprehensive design for Hamiltonian decompositions of complete $k$-uniform hypergraphs $K_n^k$. Building upon the resolution of the long-standing conjecture of the existence of Hamiltonian decompositions of complete hypergraphs, a problem that was resolved using existence-based methods, our contribution goes beyond the previous explicit designs, which were confined to the specific cases of $k=2$ and $k=3$, by providing explicit designs for all $k$ and $n$ prime, allowing for a broad applicability of Hamiltonian decompositions in various settings.

Constructing Hamiltonian Decompositions of Complete $k$-Uniform Hypergraphs

TL;DR

The paper addresses the problem of partitioning the edge set of the complete -uniform hypergraph into Hamiltonian cycles, extending previous existence results to explicit designs for all with prime and . It introduces a generator-based framework that maps each -subset to a Hamiltonian cycle via a generator vector , a representative vector , and the cycle , with a cycle period equal to the number of vertices when conditions hold (e.g., ). By constructing a comprehensive set of generators , determined from parameters and , the authors ensure that the corresponding vector sets partition into disjoint Hamiltonian cycles, providing an explicit Hamiltonian decomposition. The work delivers concrete constructions for all with prime , with a detailed example for and discussion of practical implications for distributed systems, coded caching, routing, and fault tolerance.

Abstract

Motivated by the wide-ranging applications of Hamiltonian decompositions in distributed computing, coded caching, routing, resource allocation, load balancing, and fault tolerance, our work presents a comprehensive design for Hamiltonian decompositions of complete -uniform hypergraphs . Building upon the resolution of the long-standing conjecture of the existence of Hamiltonian decompositions of complete hypergraphs, a problem that was resolved using existence-based methods, our contribution goes beyond the previous explicit designs, which were confined to the specific cases of and , by providing explicit designs for all and prime, allowing for a broad applicability of Hamiltonian decompositions in various settings.

Paper Structure

This paper contains 14 sections, 3 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and Applications
  3. Prior Work
  4. Notations
  5. Preliminaries
  6. Definitions
  7. Adopted ordering
  8. Examples
  9. Designing the Hamiltonian Decomposition
  10. Generator-Based Framework
  11. Designing Generators
  12. Proof of Hamiltonian Decomposition
  13. Example of a New Construction
  14. Conclusions

Key Result

Proposition 1

If $n$ and $k$ are coprime, then the generator $\underline{g} = [g_1, g_2, \dots, g_k]$ has period $n$.

Figures (1)

  • Figure 1: A configuration of the representative $\underline{r}_{\underline{g}} = [r_1, r_2, \dots, r_k]$. The corresponding generator $\underline{g}$ indicates the difference between consecutive entries of $\underline{r}_{\underline{g}}$, meaning that $g_1=r_2-r_1, \ \dots,\ g_{k-1}=r_k-r_{k-1}$. The period indicates the minimum number of clockwise cyclic shifts required to return to the starting configuration.

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Remark 1