Schülke’s Pósa-type criterion conjecture for tight Hamiltonian cycles

Let $k\ge 3$ be fixed and let $H$ be a $k$ -uniform hypergraph on $n$ vertices. For each $(k-2)$ -set $S\subseteq V(H)$ , let $L_H(S)$ be the link graph of $S$ , whose vertices are $V(H)\setminus S$ and whose edges are the pairs $\{x,y\}$ such that $S\cup\{x,y\}\in E(H)$ . Schülke’s conjecture asks whether the stated Pósa-type degree condition on the link graphs—equivalently, a condition combining the codegrees $\deg_H(S\cup\{x\})=\deg_{L_H(S)}(x)$ with the numbers of low-degree vertices in each $L_H(S)$ —guarantees that $H$ contains a tight Hamiltonian cycle. The supplied sources do not state the exact inequality defining this condition.

References

Primary source

arXiv

Additional references

A Chvátal-type codegree condition for Hamiltonian cycles in k-uniform hypergraphs — arXiv

extremal-combinatorics hypergraph-theory

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A new unrefereed preprint claims to settle the conjectured test for tight Hamiltonian cycles, but its result has not been independently verified.

The problem concerns Schülke’s Pósa-type criterion conjecture for tight Hamiltonian cycles in uniform hypergraphs. The conjecture proposes a condition combining codegrees with degrees in link graphs.

September 2026 claimed resolution

A September 2026 arXiv preprint claims a Chvátal-type Hamilton-cycle criterion for uniform hypergraphs, allowing small codegrees to be compensated by large link-graph degrees under an asymptotic condition. This appears to address, and potentially settle, Schülke’s conjectured criterion, but the preprint is unrefereed and the proof remains unverified.

Current status (as of September 2026): A preprint claims to prove the conjectured criterion, but independent verification is absent.

Sources

No solutions have been posted yet.