Packing Hamilton Cycles in Random and Pseudo-Random Hypergraphs

TL;DR

This paper demonstrates that in certain random and pseudo-random hypergraphs, most edges can be decomposed into edge-disjoint Hamilton cycles of specified types, extending understanding of hypergraph cycle packings.

Contribution

It establishes probabilistic and pseudo-random conditions under which hypergraphs can be decomposed into Hamilton cycles of a given type, generalizing previous results.

Findings

Almost all edges in certain random hypergraphs can be decomposed into Hamilton cycles.

Pseudo-random hypergraphs can be similarly decomposed under specific conditions.

Results include packing edges into perfect matchings for the case when =k.

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.