Berge Hamilton Cycles in Hypergraphs

• Berge Hamilton cycles in hypergraphs are cycles that alternate between distinct vertices and hyperedges, visiting each vertex exactly once.
• Dirac- and Pósa-type theorems extend to these cycles, establishing sharp minimum degree thresholds and enabling robust decomposition methods.
• Random hypergraph models exhibit sudden phase transitions for Berge Hamilton cycles, with precise threshold probabilities and hitting-time phenomena.

A Berge Hamilton cycle in a hypergraph is a cycle that traverses each vertex exactly once, with the structure defined such that the cycle alternates between distinct vertices and distinct hyperedges, and every successive pair of vertices in the sequence is contained in the associated hyperedge. The Berge notion of cycles generalizes classical Hamiltonicity in graphs to the hypergraph context and enables robust generalizations of Dirac- and Pósa-type theorems, extremal decomposition problems, and random process thresholds. These cycles play a central role in hypergraph Hamiltonicity theory, underpinning questions of existence, structure, extremal bounds, and probabilistic thresholds.

1. Definitions and Fundamental Properties

Let $H=(V,E)$ be a hypergraph (possibly uniform or non-uniform).

• A Berge cycle of length $\ell$ in $H$ is an alternating sequence of distinct vertices $v_1, e_1, v_2, e_2, \dots, v_\ell, e_\ell$ (indices modulo $\ell$) such that for all $i$, $\{v_i, v_{i+1}\} \subseteq e_i$ and each $v_i$ and $e_i$ is distinct.
• A Berge Hamilton cycle is a Berge cycle of length $n=|V|$, i.e., it visits every vertex once.
• For $r$-uniform hypergraphs ($r \geq 2$), each edge $e \in E$ has cardinality exactly $r$.
• The shadow graph $\partial_2(H)$ of $H$ has the same vertex set, with two vertices adjacent if they appear together in any hyperedge.

A critical distinction is made between weak Berge cycles (edges need not be distinct) and strong or classic Berge cycles (where all used edges are distinct).

2. Structural and Extremal Results

Dirac-type Theorems: Analogous to the classical Dirac bound for graphs, precise minimum vertex-degree thresholds for Hamiltonicity in $r$-uniform hypergraphs have been established. The central results for $r$-uniform hypergraphs $H$ on $n$ vertices are:

• If $r < n/2$, then

$$\delta(H) \geq \binom{\lfloor (n-1)/2 \rfloor}{r-1} + 1 \implies H \text{ contains a Hamiltonian Berge cycle}$$

• If $r \geq n/2$, then

$$\delta(H) \geq r \implies H \text{ contains a Hamiltonian Berge cycle}$$

These thresholds are sharp; extremal constructions are provided via clique amalgamations or deletion of edges in tight cycles (Kostochka et al., 2021).

Pósa-type Degree Sequences: Generalizing Pósa’s theorem, sharp bounds on the entire degree sequence suffice for Berge Hamiltonicity. For $r$-uniform $n$-vertex $H$ ($n > 2r$), the sequence is $r$-Hamiltonian if:

• $d_i > i$ for $1\le i<r$,
• $d_i > \binom{i}{r-1}$ for $r \le i \le \lfloor (n-1)/2 \rfloor$,
• $d_{(n-2)/2} > \binom{(n-2)/2}{r-1} + 1$ when $n$ is even.

Similarly, for non-uniform hypergraphs, exponential degree lower bounds suffice (Salia, 2021).

Connectedness and Long Cycles: For 2-connected $r$-uniform hypergraphs,

$$\delta(H) \geq \binom{k-1}{r-1} + 1 \Longrightarrow H \text{ contains a Berge cycle of length }\min\{2k,n\}$$

with the Hamiltonian case when $k \ge n/2$ (Kostochka et al., 2022).

3. Random Bounded and Hitting-Time Thresholds

Hamiltonian Berge cycles in random hypergraphs exhibit a phase transition at the minimum degree threshold of 2:

• In the random $r$-uniform hypergraph process, as soon as the minimum vertex-degree becomes 2 (time $T_2$), a Hamiltonian Berge cycle almost surely appears (Bal et al., 2018).
• In the binomial model $G^{(r)}(n,p)$, the threshold for Hamiltonian Berge cycles is

$$p(n) = (r-1)! \frac{ \log n + \log \log n + c_n }{ n^{r-1} }$$

with the usual double-jump behavior: probability tends to 1 if $c_n\to+\infty$, to 0 if $c_n\to-\infty$.

• In the $2$-out random $r$-graph, $G^{(r)}(n,2)$ is whp Berge Hamiltonian for $r \ge 3$; $G^{(r)}(n,1)$ is not.
• The result extends to random subgraph processes in dense host hypergraphs: For $H$ with

$$\delta_1(H) \geq (1/2^{r-1}+\varepsilon)\binom{n-1}{r-1} \quad \text{or} \quad \delta_2(H) \geq \varepsilon n^{r-2},$$

the hitting time for minimum degree 2 coincides with the appearance of a Hamiltonian Berge cycle with high probability (Im et al., 7 Dec 2025).

4. Constructions, Decompositions, and Special Hypergraph Classes

• Decompositions: The complete $r$-uniform hypergraph $K_n^{(r)}$ admits a decomposition into Hamiltonian Berge cycles whenever $n$ divides $\binom{n}{r}$ and $n, r$ are sufficiently large. The construction leverages perfect matchings in auxiliary bipartite graphs and decompositions of the complete digraph into Hamilton cycles (Kühn et al., 2014).
• $\sigma$-Hypergraphs: For $r$-uniform hypergraphs defined via a partition $\sigma$ of $r$, most $\sigma$-hypergraphs are Hamiltonian in the Berge sense under mild divisibility requirements. Moreover, such frameworks allow the study of sharper/stronger cycle structures, such as sharp cycles (only consecutive edges intersect) and $k$-intersecting Hamiltonian cycles (Zarb, 2014).

Construction/Class	Hamiltonicity Result	Reference
Complete $r$-graphs	Decomposition into Hamilton Berge cycles for $n\mid\binom{n}{r}$	(Kühn et al., 2014)
$\sigma$-hypergraphs	Most cases admit a Hamiltonian Berge cycle; also sharp cycles	(Zarb, 2014)
Covering $3$-graphs	Pancyclicity and tight extremal Lagrangian results	(Lu et al., 2019)

• Monochromatic Hamilton cycles: Every $(r-2)$-coloring of $K_n^r$, for large $n$, contains a monochromatic Hamiltonian Berge cycle; the general threshold and the $r=4$ case are resolved with explicit bounds (Omidi et al., 2014).

5. Proof Techniques and Methodological Advances

• Rotation-Extension (Pósa-type argument): Generalizes from graphs: starting from a longest path, endpoint-rotations and edge replacements yield large sets of possible endpoints, which, under degree conditions, force closure into a Hamiltonian cycle (Salia, 2021, Kostochka et al., 2021).
• Lollipop and “best-pair” approach: Max-min strategies based on maximizing the cycle length and then the associated attached path, combined with forbidden set analysis and fine interval arguments (Kostochka et al., 2021).
• Absorption Method: Especially in random/pseudorandom settings, absorber structures and sequential bridging operations are critical for establishing the appearance of Hamiltonian Berge cycles immediately at the relevant hitting time (1903.09057).
• Bipartite Incidence Graphs: Many proofs exploit the equivalence between Berge cycles and cycles in bipartite incidence graphs, allowing classical matching and expansion arguments (Kostochka et al., 2019, Kostochka et al., 2020).
• Booster Edges: In the random process, the addition of boosters (edges that close or extend cycles) is systematically analyzed, generalizing Pósa’s booster concept to hypergraph settings (Bal et al., 2018, Im et al., 7 Dec 2025).

6. Generalizations, Open Problems, and Further Directions

• Super-pancyclicity: Under certain Dirac-type degree and size conditions, a hypergraph can contain Berge cycles of arbitrary lengths (super-pancyclicity), not just Hamiltonian cycles (Kostochka et al., 2019, Lu et al., 2019).
• Spectral Conditions: The spectral radius threshold for Hamiltonicity in $r$-graphs mirrors Fiedler–Nikiforov’s inequality:

$$\lambda(\mathcal{H}) > \binom{n-2}{r-1} \implies \mathcal{H} \text{ Hamiltonian unless } K_{n-1}^r + e$$

with exact extremal examples (Brooks et al., 25 Apr 2025).

• Variations and Stronger Cycle Notions: Extensions exist for $\ell$-cycles, sharp cycles, $k$-intersecting cycles, and more, each requiring distinct divisibility and structural conditions (Zarb, 2014).
• Random Models: Beyond classical Erdős–Rényi, further work investigates sharper thresholds, resilience under edge deletions, and hitting-time phenomena in pseudorandom and sparse models (Im et al., 7 Dec 2025, 1903.09057).
• Algorithmic and Constructive Aspects: Many decomposition results are existential, but several recent works develop explicit construction algorithms and analyze complexity.
• Chvátal-type Degree Sequences: Determining necessary and sufficient conditions analogous to Chvátal’s theorem remains open (Salia, 2021).

7. Connections to Graph Hamiltonicity and Impact

The study of Berge Hamilton cycles reveals a profound interplay between hypergraph expansion, combinatorial structure, and probabilistic methods. The generalization of classical Dirac and Pósa results underscores the robustness of Hamiltonicity as a unifying concept in extremal combinatorics. Key advances, like the tight minimum-degree thresholds, sharp probabilistic phase transitions, and decomposition theorems, position Berge Hamiltonicity as a central paradigm for probing the limits of connectivity and cyclicity in large and complex combinatorial systems (Kostochka et al., 2021, Im et al., 7 Dec 2025, Kostochka et al., 2022).