Tight Hamilton cycles in random hypergraphs

TL;DR

This paper presents an algorithmic proof demonstrating the existence of tight Hamilton cycles in random hypergraphs with edge probability p=n^{-1+eps} for all eps>0, advancing understanding in probabilistic combinatorics.

Contribution

It introduces a new algorithmic approach to establish the existence of tight Hamilton cycles in random hypergraphs, extending previous probabilistic results.

Findings

Existence of tight Hamilton cycles for p=n^{-1+eps} in r-uniform hypergraphs.

Method applies to related combinatorial problems.

Partially answers open questions in random hypergraph theory.

Abstract

We give an algorithmic proof for the existence of tight Hamilton cycles in a random r-uniform hypergraph with edge probability p=n^{-1+eps} for every eps>0. This partly answers a question of Dudek and Frieze [Random Structures Algorithms], who used a second moment method to show that tight Hamilton cycles exist even for p=omega(n)/n (r>2) where omega(n) tends to infinity arbitrary slowly, and for p=(e+o(1))/n (r>3). The method we develop for proving our result applies to related problems as well.