Codegree Thresholds for Tight Hypergraph Hamilton Cycles

Definitions and mathematical statement

Let $k,n$ be integers with $n\geq k+1\geq4$, and let $H=(V,E)$ be a $k$-uniform hypergraph with $|V|=n$ and $E\subseteq\binom Vk$. For $S\in\binom V{k-1}$ define its codegree by \[d_H(S)=|\{v\in V\setminus S:S\cup\{v\}\in E\}|,\qquad\delta_{k-1}(H)=\min_{S\in\binom V{k-1}}d_H(S).\] A tight Hamiltonian cycle is a cyclic ordering $v_1,\ldots,v_n$ of all vertices such that $\{v_i,v_{i+1},\ldots,v_{i+k-1}\}\in E$ for every $i$, with indices taken modulo $n$. The conjecture asks whether \[\delta_{k-1}(H)\geq\left\lfloor\frac{n-k+3}{2}\right\rfloor\quad\Longrightarrow\quad H\text{ contains a tight Hamiltonian cycle}.\] The source quantifies over the full stated range, without a sufficiently-large-$n$ qualification, an asymptotic error, or a divisibility condition on $n$. The supplied 2026-08-21 literature triage labels the record partially solved, reports a proved $3$-uniform case and important further ranges, and says general $k$-uniform tight Hamilton-cycle thresholds remain open. It does not state the precise size hypotheses of those partial results, so they are not silently promoted to the full parameter range above.

Short English statement

Does minimum codegree at least $\lfloor(n-k+3)/2\rfloor$ force a cyclic ordering in which every $k$ consecutive vertices form an edge, for every $n\geq k+1\geq4$? The supplied assessment reports partial results without a general resolution.

Sources

• Local UnsolvedMath registry, permanent ID 30000899, source problem catalogue entry, Codegree Thresholds for Tight Hypergraph Hamilton Cycles, proposed year 2008; no individual proposer supplied.
• Combinatorics (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/01, pp. 5--78: https://doi.org/10.4171/owr/2008/01. Supplied literature label: original Oberwolfach report.
• Supplied literature label: Lang, tight Hamilton-cycle thresholds, https://doi.org/10.1112/jlms.12561.
• Supplied literature label: minimum codegree theorem, https://doi.org/10.1016/j.jcta.2015.01.004.