Powers of Hamiltonian cycles in randomly augmented graphs

TL;DR

This paper demonstrates that adding a linear number of random edges to a large minimum degree graph guarantees the existence of higher powers of Hamiltonian cycles with high probability, extending classical deterministic results.

Contribution

It shows that minimal random augmentation suffices to ensure higher powers of Hamiltonian cycles, building on classical degree conditions.

Findings

Adding O(n) random edges guarantees (k+1)-th power of Hamiltonian cycle

High probability of existence under degree condition

Extends deterministic Hamiltonian cycle results to randomized settings

Abstract

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Koml\'os, Sark\"ozy, and Szemer\'edi that for every $k\geq 1$ and sufficiently large $n$ already the minimum degree $\delta(G)\ge\tfrac{k}{k+1}n$ for an $n$-vertex graph $G$ alone suffices to ensure the existence of a $k$-th power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just $O(n)$ random edges ensures the presence of the $(k+1)$-st power of a Hamiltonian cycle with probability close to one.