Sharp thresholds for higher powers of Hamilton cycles in random graphs

TL;DR

This paper proves the precise sharp threshold for the appearance of the k-th power of Hamilton cycles in random graphs for k ≥ 4, refining previous weak threshold results using an advanced second moment method.

Contribution

It establishes the exact constant for the sharp threshold of the k-th power of Hamilton cycles in random graphs, improving upon prior weak threshold results.

Findings

Identifies the sharp threshold at p = (e/n)^{1/k} for k ≥ 4.

Refines the second moment method to precisely control subgraph contributions.

Determines the exact constant in the threshold for the k-th power of Hamilton cycles.

Abstract

For $k \geq 4$, we establish that $p = (e/n)^{1/k}$ is a sharp threshold for the existence of the $k$-th power $H$ of a Hamilton cycle in the binomial random graph model. Our proof builds upon an approach by Riordan based on the second moment method, which previously established a weak threshold for $H$. This method expresses the second moment bound through contributions of subgraphs of $H$, with two key quantities: the number of copies of each subgraph in $H$ and the subgraphs' densities. We control these two quantities more precisely by carefully restructuring Riordan's proof and treating sparse and dense subgraphs of $H$ separately. This allows us to determine the exact constant in the threshold.