Positive codegree thresholds for Hamilton cycles in hypergraphs

2025/05/16 by Richard Mycroft, Mycroft, Richard, Camila Zárate-Guerén +1 · 2 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2505.11400

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each k ≥ 3 and 1 ≤ ℓ ≤ k-1 we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton ℓ-cycle in a k-uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case ℓ = k-1 of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, Müyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.

Citations

• Positive co-degree densities and jumps
• Positive co-degree thresholds for spanning structures
• Spanning spheres in Dirac hypergraphs
• Positive co-degree density of hypergraphs
• Maximum size intersecting families of bounded minimum positive co-degree
• Maximum Size Intersecting Families of Bounded Minimum Positive Co-degree
• Minimum degree thresholds for Hamilton (k/2)-cycles in k-uniform hypergraphs
• Hamilton cycles in hypergraphs below the Dirac threshold
• Recent advances on Dirac-type problems for hypergraphs
• Hamilton cycles in graphs and hypergraphs: an extremal perspective
• Tight co-degree condition for the existence of loose Hamilton cycles in\n 3-graphs
• Some Theorems on Abstract Graphs

Cited by

• Loose Hamiltonicity
• Positive codegree thresholds for perfect matchings in hypergraphs

Related

• Localised codegree conditions for tight Hamilton cycles in 3-uniform hypergraphs
• Forbidding Hamilton cycles in uniform hypergraphs
• Tight Hamilton Cycles in Random Uniform Hypergraphs
• Minimum degree thresholds for Hamilton (ℓ,k-ℓ)-cycles in k-uniform hypergraphs
• Hamilton cycles in hypergraphs below the Dirac threshold