Sharp thresholds for higher powers of Hamilton cycles in random graphs

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.