[Paper Review] Packing Hamilton Cycles in Random and Pseudo-Random Hypergraphs

Alan Frieze, Michael Krivelevich|arXiv (Cornell University)|Mar 9, 2010

Limits and Structures in Graph Theory8 references4 citations

TL;DR

This paper establishes sufficient conditions for packing almost all edges of random and pseudo-random k-uniform hypergraphs into edge-disjoint Hamilton cycles of type ℓ, where 1 ≤ ℓ ≤ k and ℓ ≤ k ≤ 2ℓ. Using probabilistic methods and degree-based regularity conditions, the authors prove that for edge probability p ≫ log²n/n, whp H(n,p,k) contains (1−o(1))·(n choose k)·p/νℓ such cycles, extending prior results on Hamilton cycle packing in random graphs to the hypergraph setting with tight control on cycle structure and edge disjointness.

ABSTRACT

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}-E_i|=\ell$. We prove that for $\ell \le k\le 2\ell$, with high probability almost all edges of a random $k$-uniform hypergraph $H(n,p,k)$ with $p(n)\gg \log^2 n/n$ can be decomposed into edge disjoint type $\ell$ Hamilton cycles. We also provide sufficient conditions for decomposing almost all edges of a pseudo-random $k$-uniform hypergraph into type $\ell$ Hamilton cycles, for $\ell \le k\le 2\ell$. For the case $\ell=k$ these results show that almost all edges of corresponding random and pseudo-random hypergraphs can be packed into disjoint perfect matchings.

Motivation & Objective

• To extend the theory of Hamilton cycle packing from random graphs to random and pseudo-random k-uniform hypergraphs.
• To define and analyze Hamilton cycles of type ℓ in k-uniform hypergraphs, where consecutive edges intersect in exactly ℓ vertices.
• To establish sufficient conditions—based on vertex degree and co-degree regularity—for decomposing almost all edges into edge-disjoint type-ℓ Hamilton cycles.
• To provide tight bounds on the edge probability p(n) ensuring such packings whp, especially for ℓ ≥ k/2.
• To generalize prior results on Hamilton cycle packing in random graphs to the hypergraph regime, including the case ℓ = k (perfect matchings).

Proposed method

• Define a type-ℓ Hamilton cycle in a k-uniform hypergraph as a cyclic sequence of edges where each edge consists of k consecutive vertices and consecutive edges intersect in exactly ℓ vertices.
• Introduce regularity conditions (Pa–Pf) on the minimum and maximum degrees of vertex sets of size 2(k−ℓ), 2ℓ−k, etc., based on expected degrees in random hypergraphs.
• Use Chernoff bounds and martingale concentration inequalities to show that random and pseudo-random hypergraphs satisfy these regularity conditions whp.
• Construct edge-disjoint Hamilton cycles via random partitioning and labeling: partition vertex sets into parts, assign labels to edges based on inclusion in random substructures, and use regularity to guarantee Hamilton cycles in the labeled graphs.
• Apply Theorem 1 (on Hamilton cycle packing in (α,ε)-regular graphs) to the labeled subgraphs Gi to derive lower bounds on the number of edge-disjoint cycles.
• Prove that the resulting cycles are edge-disjoint and cover (1−o(1)) of the total edge set, under the stated conditions on p(n) and ǫ.

Experimental results

Research questions

• RQ1 What is the threshold edge probability p(n) for whp packing almost all edges of a random k-uniform hypergraph H(n,p,k) into edge-disjoint type-ℓ Hamilton cycles, for ℓ ≤ k ≤ 2ℓ?
• RQ2 Can sufficient conditions on the degree and co-degree distributions of a pseudo-random k-uniform hypergraph guarantee a near-complete packing of its edges into type-ℓ Hamilton cycles?
• RQ3 How does the structure of type-ℓ Hamilton cycles—defined by ℓ-sized intersections between consecutive edges—affect the feasibility and density of such packings?
• RQ4 To what extent can the techniques used for random graphs be extended to hypergraphs, particularly in the sparse regime where o(n^k) edges are present?
• RQ5 What are the tightest possible bounds on p(n) and the regularity parameter ε for such packings to hold whp?

Key findings

• For ℓ ≤ k ≤ 2ℓ and p(n) ≫ log²n/n, whp H(n,p,k) contains (1−o(1))·(n choose k)·p/νℓ edge-disjoint type-ℓ Hamilton cycles.
• The number of such cycles is asymptotically optimal, as it covers (1−o(1)) of the total edge set.
• For ℓ = k, the result implies that whp almost all edges of H(n,p,k) can be packed into edge-disjoint perfect matchings.
• The sufficient conditions for pseudo-random hypergraphs (Pa–Pf) are shown to hold whp in random hypergraphs, making the results applicable to both models.
• The method relies on random partitioning and labeling, followed by application of Theorem 1 to subgraphs Gi, ensuring Hamilton cycle existence via regularity.
• The analysis uses concentration inequalities (Chernoff bounds and martingale-type arguments) to verify that the labeled subgraphs Gi are (α,ε)-regular, enabling cycle packing.