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 establish a precise characterisation of $4$-uniform hypergraphs withminimum codegree close to $n/2$ which contain a Hamilton $2$-cycle. As animmediate corollary we identify the exact Dirac threshold for Hamilton$2$-cycles in $4$-uniform hypergraphs. Moreover, by derandomising the proof ofour characterisation we provide a polynomial-time algorithm which, given a$4$-uniform hypergraph $H$ with minimum codegree close to $n/2$, either finds aHamilton $2$-cycle in $H$ or provides a certificate that no such cycle exists.This surprising result stands in contrast to the graph setting, in which belowthe Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. Wealso consider tight Hamilton cycles in $k$-uniform hypergraphs $H$ for $k \geq3$, giving a series of reductions to show that it is NP-hard to determinewhether a $k$-uniform hypergraph $H$ with minimum degree $\delta(H) \geq\frac{1}{2}|V(H)| - O(1)$ contains a tight Hamilton cycle. It is thereforeunlikely that a similar characterisation can be obtained for tight Hamiltoncycles.

Hamilton cycles in hypergraphs below the Dirac threshold