k-Uniform Tight Cycles

• k-Uniform tight cycles are hypergraphs with vertices in a cyclic order where each edge consists of k consecutive vertices, overlapping by k-1 vertices.
• Recent advances detail their extremal, Turán-type, and decomposition properties, reducing bounds and refining partition results in monochromatic settings.
• These cycles underpin critical results in Hamiltonicity, Ramsey theory, and probabilistic thresholds, impacting algorithmic and combinatorial applications.

A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices, where every edge is formed by $k$ consecutive vertices, and consecutive edges overlap in exactly $k-1$ vertices. Tight cycles generalize the classical notion of cycles in graphs and underpin key questions in extremal, Ramsey, and decomposition theory for hypergraphs. Their structure, covering, partitioning, threshold, and extremal properties are central to modern combinatorics, with recent advances revealing deep analogies and new obstacles compared to the graph case.

1. Formal Definition, Properties, and Examples

Let $k\ge2$ and $n \ge k$. A $k$-uniform tight cycle $C_n^{(k)}$ is defined on vertex set $\{v_0, v_1, \dots, v_{n-1}\}$ (indices mod $n$), with edge set

$$E(C_n^{(k)}) = \left\{ \{v_i, v_{i+1}, \dots, v_{i+k-1}\} \;:\; i=0,1,\dots,n-1 \right\}$$

Each edge is a block of $k$ consecutive vertices; consecutive edges share $k-1$ vertices, forming a single $k$-overlapping cyclic structure. To "close up" the cycle precisely, $n$ must be divisible by $k$. Degenerate cycles of length $<k$ are permitted in covering/absorbing arguments.

Examples:

• $k=3$, $n=5$:
  • $$E = \{ \{v_0,v_1,v_2\}, \{v_1,v_2,v_3\}, \{v_2,v_3,v_4\}, \{v_3,v_4,v_0\}, \{v_4,v_0,v_1\} \}$$
• $k=4$, $n=6$:
  • $$E = \{ \{v_0,v_1,v_2,v_3\}, \{v_1,v_2,v_3,v_4\}, \dots, \{v_5,v_0,v_1,v_2\} \}$$

Each vertex participates in $k$ edges; two edges at distance $\ge2$ share fewer than $k-1$ vertices.

2. Extremal and Turán-Type Properties

For any fixed $k \ge 2$, the extremal number and Turán density of tight cycles have been the subject of foundational conjectures and recent breakthroughs.

Sós–Verstraëte Conjecture: It was conjectured $f_k(n) = \binom{n-1}{k-1}$, that is, the $k$-star is extremal among tight-cycle-free hypergraphs. Huang–Ma proved this is false for $k \ge 3$; the true $f_k(n)$ is strictly larger by a positive constant (Huang et al., 2017):

$k$	Extremal edge count $f_k(n)$
2	$n-1$ (Erdős–Gallai)
$>2$	$> \binom{n-1}{k-1}$ (Huang–Ma)

Recent work characterizes the exact Turán density for long tight cycles in certain uniformities: for example, for $4$-uniform tight cycles of long length $L$ not divisible by $4$, the Turán density is exactly $1/2$ (Sankar, 2024).

3. Covering, Partitioning, and Cycle Decomposition

For complete $k$-graphs $K_n^{(k)}$ under $r$-edge colourings, the partitioning of $V(K_n^{(k)})$ into few vertex-disjoint monochromatic tight cycles is fully resolved in theory, though optimal constants remain out of reach.

Main theorem (Bustamante et al., 2019, Bandyopadhyay et al., 2024):

For all $k\ge2$, $r\ge1$, there exists $f(k,r)$ such that every $r$-edge-coloured $K_n^{(k)}$ can be partitioned into at most $f(k,r)$ monochromatic tight cycles. The bound $f(k,r)$ is independent of $n$. The tower-type constant from regularity methods is now improved to a polynomial in $r$ (Bandyopadhyay et al., 2024):

Result	Bound	Reference
Existence	$f(k,r)$ finite	(Bustamante et al., 2019)
Quantitative	$f(k,r) = (2r)^{2^{k+4}+2^{k+8} r \log(2r)}$	(Bandyopadhyay et al., 2024)

The core proof proceeds via:

• Construction of large monochromatic "crowns" (absorbers)
• Greedy almost-covering by cycles
• Absorbing residual vertices using short connecting monochromatic tight paths (guaranteed by Regular Slice Lemma)
• Absorption lemma: $k$-partite links ensure complete covering.

Recent milestone: For $r=2$, $k=4$ tight cycles, two vertex-disjoint monochromatic cycles suffice to cover all but $o(n)$ vertices (Lo et al., 2020). For general $k$, $k$ cycles suffice for almost-covering (Lo et al., 2023).

4. Degree Conditions and Hamiltonicity

For existence and packing of tight Hamilton cycles, Dirac-type thresholds have been determined in several regimes. Write $\delta_{k-1}(H)$ for minimum codegree.

• Hamiltonicity (tight cycles): For $n\gg1$, $\delta_{k-1}(H)\ge n/2 + o(n)$ implies existence of a tight Hamilton cycle (Glock et al., 2019); the bound is best possible (Wang et al., 2023). For non-$k$-partite tight cycles, there are extra barriers depending on divisibility, yielding higher thresholds, e.g., for $s \not\equiv 0 \pmod k$ (Han et al., 2017).
• Fractional decompositions: Any $k$-graph with $\delta_{k-1}(H)\ge(1/2+\varepsilon)n$ admits a fractional decomposition into tight cycles of prescribed length, for all $\ell \ge \ell_0$ (Joos et al., 2021).
• Exact tiling: For large $s$ (e.g., $s\ge5k^2$, $s \not\equiv 0 \pmod k$), a perfect tiling by vertex-disjoint tight cycles requires $\delta_{k-1}(H)\ge (1/2 + 1/(2s) + o(1))n$ (Han et al., 2017).
• Cycle decomposition: For sufficiently large $n$, $k$-graphs with $\delta_{k-1}(H)\ge(2/3+o(1))n$ admit a full tight cycle decomposition, subject to divisibility (Lo et al., 2022).

5. Ramsey Theory for Tight Cycles

The Ramsey number $R(C^{(k)}_{kn})$ for tight cycles in $k$-uniform hypergraphs captures key analogies to the graph case.

Key result (Lo et al., 2021, Pfenninger, 2024):

For tight cycles of length $kn$, $R(C^{(k)}_{kn}) = (k+1+o(1))n$ as $n\to\infty$, for all $k \ge 3$. For $k=4$, the exact asymptotics are $R(C^{(4)}_{4n}) = (5 + o(1))n$.

$k$ Uniformity	Ramsey Number $R(C^{(k)}_{kn})$
$k=3$	$(4+o(1))n$
$k=4$	$(5+o(1))n$
General $k$	$(k+1+o(1))n$

For other residue classes $i \pmod k$ of the cycle length, bounds depend intricately on divisibility and parity; new gadgets (blueprints, small tight cycles) are required for the construction of long monochromatic cycles.

6. Probabilistic Results and Random Tight Cycles

The sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs $H^k_{n,p}$ is $p^* = e/n$ for $k \ge 4$, $p^* = 1/n$ for $k = 3$ (Dudek et al., 2011).

• Threshold for tight Hamilton cycles: $p^* = e/n$ for $k \ge 4$.
• Decomposition: $(\varepsilon,p)$-regular $k$-graphs admit an almost complete edge decomposition into edge-disjoint tight Hamilton cycles when $p \gg (\log n / n)^{1/(2k)}$ (Bal et al., 2011).

The proof combines first and second moment arguments applied to embed tight cycles as labelings (cycles in permutations), and iterative reductions to digraph decompositions.

7. Advanced Constructions, Stability, and Open Problems

Recent research delivers deep algebraic and combinatorial characterizations for tight-cycle extremal structures, particularly in high uniformity ($k=4$):

• Homomorphic avoidance: For $r$-uniform tight cycles $C_L^{(r)}$, any extremal construction is characterized by the avoidance of "colored tight walks" (group-theoretic approach), implying a stable extremal structure (e.g., oddly-bipartite graphs in $r=4$) and Turán density $1/2$ for sufficiently long cycles not divisible by $r$ (Sankar, 2024).
• Codegree Turán density: For $k$-uniform tight cycles $C^k_\ell$ with $k\nmid \ell$, the codegree Turán density is $\gamma(C^k_\ell)=1/3$ for all prime $k$ and large enough $\ell$ (Ma et al., 28 Dec 2025). For $\gcd(k,\ell)=1$, density is determined on a set of $\ell$ of natural density $\varphi(k)/k$.

Open problems:

• Exact determination of $f(k,r)$ for the monochromatic cycle partition function.
• Tight bounds for degree and codegree thresholds for cycle decompositions.
• Extensions to twisted tight cycles, cycles minus one edge (e.g., extremal density for $C_\ell^{k-}$).
• Algorithmic construction and complexity of tight cycle decompositions at threshold densities.
• Stability and uniqueness of extremal configurations in non-partite settings, parity-induced barriers, and general residue classes.

References to Key Papers

• Gyárfás’s cycle partition theorem: (Bustamante et al., 2019)
• Partitioning with polynomial bounds: (Bandyopadhyay et al., 2024)
• Asymptotics for Ramsey numbers: (Lo et al., 2021, Pfenninger, 2024)
• Dirac-type degree thresholds: (Glock et al., 2019, Wang et al., 2023, Lang et al., 2020, Han et al., 2017)
• Turán-type extremal results: (Huang et al., 2017, Ma et al., 28 Dec 2025, Sankar, 2024)
• Regularity and dense cycle embedding: (Allen et al., 2014)
• Probabilistic thresholds: (Dudek et al., 2011, Bal et al., 2011)
• Fractional cycle decompositions: (Joos et al., 2021)
• Cycle decompositions, Euler tours: (Lo et al., 2022)

The theory of $k$-uniform tight cycles continues to reveal new intersections between extremal combinatorics, algebraic structures, and probabilistic methods, with sharp covering, decomposition, and Ramsey-type results at its center.