Packing hamilton cycles in random and pseudo‐random hypergraphs

Alan Frieze,Michael Krivelevich

doi:10.1002/rsa.20396

Alan Frieze, Michael Krivelevich

Open Access

https://doi.org/10.1002/rsa.20396

Copy DOI

Abstract

AbstractWe say that a k ‐uniform hypergraph C is a Hamilton cycle of type ℓ, for some 1 ≤ ℓ ≤ 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 Ei‐1,Ei in C (in the natural ordering of the edges) we have |Ei‐1 / Ei| = ℓ. We prove that for k/2 < ℓ ≤ k, with high probability almost all edges of the random k ‐uniform hypergraph H(n,p,k) with p(n) ≫ log 2n/n can be decomposed into edge‐disjoint type ℓ Hamilton cycles. A slightly weaker result is given for ℓ = k/2. We also provide sufficient conditions for decomposing almost all edges of a pseudo‐random k ‐uniform hypergraph into type ℓ Hamilton cycles, for k/2 ≤ ℓ ≤ k. For the case ℓ = k these results show that almost all edges of corresponding random and pseudo‐random hypergraphs can be packed with disjoint perfect matchings. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Highlights

The subject of Hamilton graphs and Hamiltonicity-related problems is undoubtedly one of the most central in Graph Theory, with a great many deep and beautiful results obtained
For the case = k these results show that almost all edges of corresponding random and pseudo-random hypergraphs can be packed with disjoint perfect matchings
Hamiltonicity problems occupy a place of honor in the theory of random graphs too, the reader can consult the monographs of Bollobas [4] and of Janson, Luczak and Rucinski [14] for an account of some of the most important results related to Hamilton cycles in random graphs

Summary

Introduction

The subject of Hamilton graphs and Hamiltonicity-related problems is undoubtedly one of the most central in Graph Theory, with a great many deep and beautiful results obtained. H contains a collection of (1 − 4 1/3)m/νk edge-disjoint perfect matchings These are the first results of any significance on packing Hamilton cycles in random and pseudo-random hypergraphs. An interesting point of reference for our theorems is results about perfect decompositions of the edge set of a complete k-uniform hypergraph Knk into Hamilton cycles of various types (assuming some natural divisibility conditions). We can whp reduce our problem to showing that with probability 1 − o(n−3) the random bipartite graph Kν2,ν2,p0 satisfies has a family of n0 edge disjoint perfect matchings. This is an immediate corollary of Lemma 1

A general scheme

Random hypergraphs

Pseudo-random hypergraphs

Concluding remarks