Sharp thresholds for higher powers of Hamilton cycles in random graphs

TL;DR

This paper pinpoints the sharp threshold for the existence of the $k$-th power of a Hamilton cycle in $G(n,p)$ for all $k\ge 4$, proving $p^*=(e/n)^{1/k}$ is the precise threshold. The authors extend Riordan's second-moment framework by decomposing the variance into contributions of subgraphs of the target $H$ and crucially separating sparse and dense subgraphs to tighten the bounds and determine the exact constant. A central innovation is the detailed analysis of the densest subgraphs of $H$, showing that $H_s$ are the densest on $s$ vertices and that the density increases with $s$, culminating in $H$ being the densest connected subgraph. By bounding the contributions from sparse completions and dense completions with refined embeddings-count arguments and tail bounds, the paper closes the gap between the first-moment lower bound and the second-moment upper bound, delivering a complete sharp-threshold picture with the correct constant. The results cement the understanding of high-power Hamiltonian structures in random graphs and inform future approaches to related spanning subgraph thresholds.

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.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Lemma 3: Lemma 4.2 in riordan2000
• Lemma 4
• Lemma 5
• Lemma 6
• Proposition 7
• Proposition 8
• Proposition 9
• proof : Proof of Theorem \ref{['thm:main_uniform']}