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

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 Kühn 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 Kühn 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.