English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We investigate the existence of powers of Hamiltonian cycles in graphs withlarge minimum degree to which some additional edges have been added in a randommanner. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and forany $\alpha>\frac{k}{k+1}$ we show that adding $O(n^{2-2/\ell})$ random edgesto an $n$-vertex graph $G$ with minimum degree at least $\alpha n$ yields, withprobability close to one, the existence of the $(k\ell+r)$-th power of aHamiltonian cycle. In particular, for $r=1$ and $\ell=2$ this implies thatadding $O(n)$ random edges to such a graph $G$ already ensures the $(2k+1)$-stpower of a Hamiltonian cycle (proved independently by Nenadov and Truji\'c). Inthis instance and for several other choices of $k$, $\ell$, and $r$ we can showthat our result is asymptotically optimal.

High powers of Hamiltonian cycles in randomly augmented graphs