Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

TL;DR

This paper establishes the threshold probability for the appearance of the square of a Hamilton cycle in random graphs and provides randomized algorithms for finding such cycles and tight Hamilton cycles in random hypergraphs.

Contribution

It determines the threshold for the square of a Hamilton cycle in random graphs up to a logarithmic factor and introduces a new Connecting Lemma with algorithms for cycle detection.

Findings

Threshold for the square of Hamilton cycle in G(n,p) is established.

Provides randomized quasi-polynomial algorithms for cycle detection.

Introduces a new Connecting Lemma based on Janson's inequality.

Abstract

We show that for every $k \in \mathbb{N}$ there exists $C > 0$ such that if $p^k \ge C \log^8 n / n$ then asymptotically almost surely the random graph $G_{n,p}$ contains the $k$\textsuperscript{th} power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of K\"uhn and Osthus. Moreover, our proof provides a randomized quasi-polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi-polynomial algorithm for finding a tight Hamilton cycle in the random $k$-uniform hypergraph $G_{n,p}^{(k)}$ for $p \ge C \log^8 n/ n$. The proofs are based on the absorbing method and follow the strategy of K\"uhn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of $p$. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.