Powers of Hamiltonian cycles in randomly augmented Dirac graphs -- the complete collection

TL;DR

This paper determines the probability threshold at which randomly augmented Dirac graphs almost surely contain the m-th power of a Hamiltonian cycle, extending understanding of Hamiltonian properties in random graph models.

Contribution

It provides precise estimates for the threshold probability in randomly augmented Dirac graphs to contain the m-th power of a Hamiltonian cycle, a significant extension of existing results.

Findings

Threshold probability for Hamiltonian cycle powers identified

Results apply to all Dirac graphs with minimum degree > (1/2 + ε)n

Enhances understanding of Hamiltonian properties in random augmentations

Abstract

We study the powers of Hamiltonian cycles in randomly augmented Dirac graphs, that is, $n$-vertex graphs $G$ with minimum degree at least $(1/2+\varepsilon)n$ to which some random edges are added. For any Dirac graph and every integer $m\ge2$, we accurately estimate the threshold probability $p=p(n)$ for the event that the random augmentation $G\cup G(n,p)$ contains the $m$-th power of a Hamiltonian cycle.