Hamilton Cycles in Dirac Graphs

• The paper establishes that Dirac graphs, with minimum degree at least n/2, guarantee a Hamilton cycle even under local incompatibility constraints.
• It employs advanced proof techniques such as absorption, almost-covering, and connecting methods alongside FPT algorithms to construct compatible Hamilton cycles.
• The study confirms tight minimum degree thresholds and utilizes extremal constructions to reveal resilient properties and algorithmic potential in Hamiltonian graph structures.

A Hamilton cycle in a Dirac graph is a spanning cycle that traverses every vertex exactly once in a graph whose minimum degree is at least half the number of vertices. The paper of Hamilton cycles in Dirac graphs has evolved from Dirac's theorem to rich structural, algorithmic, and robustness results, with recent work revealing their behavior under local constraints and forbidden patterns, as well as their generative and extremal properties.

1. Dirac’s Theorem and Hamiltonicity in Dense Graphs

Dirac's theorem (1952) states that any graph $G$ on $n\geq3$ vertices with minimum degree $\delta(G)\geq n/2$ contains a Hamilton cycle. Such graphs are called Dirac graphs. This minimum degree condition is both necessary and sufficient, tightly characterizing the emergence of a Hamiltonian cycle in dense graphs (Krivelevich et al., 2014, Jansen et al., 2019).

2. Incompatibility Systems and Compatible Hamilton Cycles

To formalize robustness under local constraints, Krivelevich, Lee, and Sudakov introduced the concept of incompatibility systems (Krivelevich et al., 2014). For $G=(V,E)$, an incompatibility system $\mathcal{F} = \{F_v\}_{v\in V}$ assigns to each vertex $v$ a collection of unordered pairs of incident edges: $$F_v \subseteq \left\{ \{e,e'\}: e\neq e'\in E,\ e\cap e'=\{v\} \right\}.$$ A $\Delta$-bounded incompatibility system satisfies that for any vertex $v$ and edge $e$ incident to $v$, at most $\Delta$ pairs in $F_v$ contain $e$. A cycle $C$ is compatible with $\mathcal{F}$ if for every consecutive pair of edges $e,e'$ in $C$ at $v$, $\{e,e'\}\notin F_v$.

The main result establishes that for some universal $\mu>0$, any Dirac graph with a $\mu n$-bounded incompatibility system contains a compatible Hamilton cycle, thus resolving a conjecture of Häggkvist from 1988 (Krivelevich et al., 2014).

For higher powers, Cheng, Hu, and Yang generalized this: For every $\gamma>0$ and $k\in\mathbb{N}$, there exists $\mu>0$ such that for sufficiently large $n$, if $G$ has minimum degree $\delta(G) \ge (\frac{k}{k+1}+\gamma)n$ and $\mathcal{F}$ is $\mu n$-bounded, then $G$ contains a compatible $k$-th power of a Hamilton cycle (Cheng et al., 2022). This demonstrates Dirac-type thresholds persist under mild local incompatibility.

3. Proof Techniques: Absorbing, Covering, and Extension in the Presence of Incompatibilities

The construction of compatible Hamilton cycles in Dirac graphs under incompatibility systems utilizes the absorbing method (Cheng et al., 2022). The method proceeds as follows:

• Absorption: For each $v$, build many small "absorbers"—$k$-th-power paths that can absorb $v$ while preservring compatibility. A random reservoir $R$ is selected to house absorbers for efficient absorption.
• Almost-covering: Use Szemerédi’s Regularity Lemma and a Hajnal–Szemerédi-style tiling to cover $G\setminus R$ with very long compatible $k$-th-power paths, leaving only $o(n)$ uncovered vertices.
• Connecting: Employ a robust connecting lemma to join these paths through $R$ with compatible short connectors, forming a spanning compatible structure, to which leftover vertices are ultimately inserted via absorbers.

A key ingredient is a counting and extension lemma for embedding compatible small cliques in super-regular pairs, and a random greedy selection ensuring sufficient absorbers and mates for connectivity between path ends (Cheng et al., 2022).

4. Extremal Constructions and Tightness

The minimum degree threshold $(k/(k+1) + \gamma)n$ for the $k$-th power of a compatible Hamilton cycle cannot be lowered (Cheng et al., 2022). A tight construction employs a complete $(k+1)$-partite graph with a specific partition that nearly meets the degree threshold but, by encoding incompatibility within parts, excludes compatible $k$-th powers of Hamilton cycles. Pigeonhole arguments then establish the necessity of the additional $\gamma n$ buffer.

For $k=1$, this recovers the (now robust) Dirac threshold; for larger $k$, it matches the sharp threshold from the Pósa–Seymour theorem (Komlós–Sárközy–Szemerédi), illustrating the optimality of the result under local robustness constraints.

5. Algorithmic and Structural Extensions

Relaxations of Dirac’s theorem explore Hamiltonicity when the minimum degree is slightly below $n/2$ or with only near-uniformity among degrees. Fixed-parameter tractable (FPT) algorithms have been developed for the Hamiltonicity problem when:

• At least $n-k$ vertices have degree $\geq n/2$ ("almost-Dirac"),
• All vertices have degree at least $n/2-k$ ("near-Dirac"), with running times $O(c^k n^{O(1)})$ (Jansen et al., 2019). Techniques include kernelization, the Bondy–Chvátal closure, and dynamic programming with color-coding.

Compatible Hamiltonicity has likewise been conjectured and partially solved for asymptotically sparser regimes and different types of sparse obstructions. The existence of compatible Hamilton cycles in random graphs and pseudorandom settings, and the extension to packing and enumeration, are active topics of investigation.

6. Robustness, Resilience, and Further Directions

The robustness of Hamiltonicity in Dirac graphs extends to the resilience of structures under removal or coloring constraints:

• Rainbow Hamilton cycles: For $\mu n$-bounded colorings, every Dirac graph contains a rainbow Hamilton cycle if $\mu$ is below $1/8$ (Coulson et al., 2018). The proof blends switchings, local lemma arguments, and regularity/blow-up techniques.
• Hamilton-generated cycle spaces: In Dirac and near-Dirac graphs with odd $n$, every cycle can be expressed as a symmetric difference of Hamilton cycles (Hamilton-generated property), provided $\delta(G)\geq (n-1)/2$ and $G$ is Hamilton-connected, with the proof using the parity-switcher method (Hou et al., 20 Mar 2025, Christoph et al., 2 Feb 2024).
• Connectivity preservation: Results show that in $k$-connected Dirac graphs, there exist edge-disjoint Hamilton cycles that, when removed, preserve $k$-connectivity of the residual graph, with sharp $O(k)$ dependence in $n$ (Hasunuma, 2023).
• Population and packing of Hamilton cycles: Dirac graphs always contain exponentially many Hamilton cycles, as well as Hamilton cycle transversals in families of Dirac graphs, supporting robustness in both single and multi-graph settings (Anastos et al., 2023).
• Bipartite holes and extremal barriers: Hamiltonicity is guaranteed if $\delta(G)\geq\widetilde\alpha(G)$, where $\widetilde\alpha(G)$ is the bipartite-hole-number (maximum for which all $(s,t)$-bipartite holes exist), extending Dirac’s threshold via an extremal combinatorial-geometric parameter (McDiarmid et al., 2016).

7. Open Problems and Thresholds

Many open questions remain regarding the optimal constants for local incompatibility bounds, extensions to sparser graphs, and generalized transition systems. In particular, the precise threshold $\mu^*$ for incompatibility systems to permit compatible Hamilton cycles in Dirac graphs is unknown, with existing bounds between $10^{-16}$ and $1/4$ (Krivelevich et al., 2014). Analogues in directed graphs, hypergraphs, random and pseudorandom models, and colored or partitioned edge assignments offer significant expansion potential for the theory.

The collective advances outlined elevate the Dirac graph paradigm from existence to robust, enumerative, structural, and algorithmic realms, establishing Dirac-type minimum degree not merely as an existence threshold but as a locus of rich, resilient Hamiltonian behavior under a wide spectrum of constraints and enhancements.