Dirac subgraphs of powers of cycles are Hamiltonian

Abstract: We show that, for every $\varepsilon>0$ and all sufficiently large $k$, any spanning subgraph of the $k$th power of a cycle with minimum degree at least $(1+\varepsilon)k$ contains a Hamilton cycle. This asymptotically settles a conjecture of Espuny Díaz, Lichev, and Wesolek.

• The paper demonstrates that every spanning subgraph of Cₙᵏ with minimum degree at least (1+ε)k is Hamiltonian for large k.
• It employs an absorption-based proof with randomized partitioning and matching techniques to ensure robust connectivity and path extension.
• The work generalizes Dirac’s theorem to locally dense yet globally sparse cycle power graphs, paving the way for further research on Hamiltonicity thresholds.

Hamiltonicity in Dirac Subgraphs of Cycle Powers

Introduction and Main Results

The paper "Dirac subgraphs of powers of cycles are Hamiltonian" (2606.07471) addresses the minimum degree threshold for Hamiltonicity in spanning subgraphs of the $k$th power of the cycle, $C_n^k$. The core result is that for every $\alpha>0$ and sufficiently large $k$, any spanning subgraph of $C_n^k$ with minimum degree at least $(1+\alpha)k$ is Hamiltonian. This asymptotically resolves a conjecture by Espuny D\'iaz, Lichev, and Wesolek that postulated that any subgraph of $C_n^k$ with minimum degree at least $k+1$ is Hamiltonian.

This work generalizes Dirac's theorem—originally established for complete graphs—and situates $C_n^k$ as a locally dense but globally sparse graph class displaying Dirac-type resilience for Hamiltonicity. Notably, the asymptotic nature of the main theorem introduces a minor gap from the original conjecture with regard to the minimum degree lower bound, leaving the exact threshold open for further investigation.

Technical Framework and Proof Methodology

The methodology hinges on the concept of local resilience, which, for Hamiltonicity, corresponds to the supremum $\beta$ for which all $C_n^k$0-subgraphs of a host graph retain Hamilton cycles. The case $C_n^k$1 is critical, as deleting more than half the incident edges at each vertex of a regular graph can disconnect it, precluding Hamiltonicity; thus, achieving Hamiltonicity for all $C_n^k$2-subgraphs reflects maximal robust connectivity, as in Dirac’s theorem.

$C_n^k$3 (the $C_n^k$4th power of a cycle) is interpreted as a 1-dimensional geometric analogue of $C_n^k$5, and its local structure permits high min-degree spanning subgraphs to inherit significant connectivity properties. The paper advances an absorption-based proof—nowadays standard in Hamiltonicity and perfect-matching thresholds in both random and pseudorandom graphs. The absorber construction is based around beehive gadgets, which can sequentially incorporate leftover vertices into constructed long paths without loss of control on local structure, providing a robust mechanism for spanning path extension.

A key technical contribution is the path decomposition lemma: for sufficiently large $C_n^k$6 and $C_n^k$7, every $C_n^k$8-Dirac subgraph of $C_n^k$9 contains a spanning linear forest with endpoints forming a set that is locally sparse in every neighborhood of size $\alpha>0$0, which is essential for the later path connection step. This decomposition, achieved through randomized partitioning and iterative application of matching-based coverings, is pivotal for controlling both the number and distribution of path endpoints, ensuring effective absorption and path-joining in the later phases of the construction.

Connecting these endpoint pairs into a single Hamiltonian cycle poses further challenges due to the lack of global expansion in $\alpha>0$1. A sophisticated interval covering and matching algorithm is utilized, with careful accounting to avoid endpoint concentration and to guarantee all connection paths remain vertex-disjoint and predominantly local.

Strength of Results and Comparisons

Major numerical thresholds:

• For all $\alpha>0$2 and sufficiently large $\alpha>0$3, every spanning subgraph of $\alpha>0$4 with minimum degree at least $\alpha>0$5 is Hamiltonian.
• The construction shows resilience up to the $\alpha>0$6 local density threshold, bringing the result in line with classic Dirac/Chvátal-type conditions for dense graphs, but in a context where the host is globally sparse but locally dense.

The paper presents additional extremal and Turán-type results for clique containment in $\alpha>0$7. Specifically, it determines the asymptotically sharp minimum degree threshold for $\alpha>0$8 subgraph containment and computes the extremal density for triangle- and clique-free subgraphs. Notably, for fixed $\alpha>0$9, the extremal density of $k$0-free subgraphs in $k$1 significantly exceeds the classical Turán density for the complete graph, indicating that the local 1D structure of $k$2 allows for denser $k$3-free subgraphs than would be admissible in $k$4.

Explicit bounds are established, for example:

• For triangles ($k$5), $k$6 as $k$7, which strictly surpasses the classical Turán bound of $k$8 for $k$9-free subgraphs in $C_n^k$0.

Broader Implications and Open Problems

The paper’s results underscore the high local resilience of $C_n^k$1 for Hamiltonicity, paralleling that of $C_n^k$2 and robust random graphs, despite different global properties. The absorption, matching, and decomposition arguments disclose a nuanced structure theory for Hamiltonicity thresholds in locally dense and globally sparse graphs—motivating investigation in higher-dimensional geometric graphs, Cayley graphs over larger groups, or other natural pseudorandom models.

Key open problems are:

• Proving the conjecture in its exact form for all $C_n^k$3 (i.e., establishing that all $C_n^k$4-Dirac subgraphs are Hamiltonian for all sufficiently large $C_n^k$5).
• Determining the precise value of the extremal density $C_n^k$6 for all $C_n^k$7.
• Extending the result to powers of higher-dimensional cycles and various geometric random graph models, noting that the naive generalization to $C_n^k$8 for $C_n^k$9 fails.

The appendices also provide nontrivial counterexamples, constructed via explicit geometric configurations, showing failure of strong local resilience of Hamiltonicity in certain random geometric graphs in dimension $(1+\alpha)k$0.

Conclusion

This paper delivers a comprehensive asymptotic confirmation of a Dirac-type local resilience threshold for Hamiltonicity in spanning subgraphs of the $(1+\alpha)k$1th power of the cycle. In doing so, it extends the scope of classical Hamiltonicity theorems to locally dense geometric and algebraic host graphs, connects absorption and matching techniques to the geometry of $(1+\alpha)k$2, and opens fruitful directions for extremal, probabilistic, and geometric combinatorics. The theoretical framework established provides a robust toolkit for analyzing the interplay between local density, global structure, and combinatorial spanning properties in sparse, structured graph families.