A cubic refinement of Jackson's Chvátal--Erdős condition for Hamilton cycles in digraphs

Abstract: For a digraph $D$, let $\aTwo(D)$ be the largest size of a vertex set no two of whose vertices lie in a common directed $2$-cycle. Let $f_2(a)$ be the least integer $K$ such that every $K$-connected digraph $D$ with $\aTwo(D)\leq a$ has a Hamilton cycle. In 1987, Jackson proved that $f_2(a)\leq 2<sup>a(a+2)!$ and asked for better bounds, noting that a linear bound might be possible. Kühn and Osthus later observed that even a polynomial bound would be interesting. In this short note, we prove the polynomial bound $f_2(a)\leq 2a<sup>3+2$.

• The paper establishes a cubic upper bound (2a³+2) on the connectivity threshold f₂(a) required for Hamiltonicity in digraphs with bounded 2-cycle independence.
• It employs combinatorial decompositions, path covers, and contraction lemmas to piece together symmetric digraph components into a Hamilton cycle.
• The refinement significantly improves on Jackson’s factorial bound, opening new avenues for algorithmic and theoretical advances in extremal digraph theory.

A Cubic Upper Bound for Hamiltonicity via a Jackson’s Chvátal–Erdős Analogue in Digraphs

Introduction and Motivation

Hamiltonicity in digraphs under vertex-connectivity and forbidden substructure constraints represents a central concern in extremal graph theory. The Chvátal–Erdős theorem provides a classical condition for Hamiltonicity in undirected graphs: if the vertex-connectivity $\kappa(G)$ is at least the independence number $\alpha(G)$, then $G$ is Hamiltonian. The extension to digraphs, initially undertaken by Jackson, substitutes the undirected independence number with a parameter $(D)$, the maximal number of vertices in $D$ no two of which belong to a common directed 2-cycle.

Jackson proved the existence of a connectivity threshold, denoted $f_2(a)$, ensuring Hamiltonicity for digraphs with $(D)\leq a$, providing an upper bound of $f_2(a)\leq 2^a(a+2)!$ and conjecturing that a linear or at least a polynomial bound might hold. Previous attempts to close the gap between this exponential upper bound and the trivial lower bound $f_2(a)\geq a$ (with exact values known for small $a$) have not produced polynomially bounded results. Addressing this significant gap, the paper establishes a cubic upper bound on $\alpha(G)$0, showing that $\alpha(G)$1.

Key Definitions and Framework

Let $\alpha(G)$2 be a finite simple digraph. Denote by:

• $\alpha(G)$3: The undirected 2-cycle graph of $\alpha(G)$4, where an edge $\alpha(G)$5 exists iff both $\alpha(G)$6 and $\alpha(G)$7 are arcs in $\alpha(G)$8.
• $\alpha(G)$9: The maximal size of a subset no two of which are contained in a common 2-cycle in $G$0.
• $G$1-connected: $G$2 is $G$3-connected if $G$4 and $G$5 is strongly connected for every vertex set $G$6 of size $G$7.
• $G$8: The minimal integer $G$9 such that every $(D)$0-connected digraph $(D)$1 with $(D)$2 is Hamiltonian.

Main Results

The paper proves the following principal theorem:

For every integer $(D)$3, $(D)$4. That is, every $(D)$5-connected digraph $(D)$6 with $(D)$7 is Hamiltonian.

This is the first known polynomial (cubic) upper bound for $(D)$8. The new bound narrows the gap with the lower bound (trivially $(D)$9), drastically improving on Jackson’s factorial upper bound.

Technical Contributions and Proof Approach

The argument is rooted in several combinatorial decompositions and the application of classical theorems:

1. Structural Decomposition: The undirected 2-cycle graph $D$0 is decomposed into a bounded number (at most $D$1) of induced subgraphs, each either complete or with connectivity exceeding a certain threshold $D$2, plus an exceptional set $D$3 of bounded size.
2. Symmetric Digraph Pieces: These subgraphs correspond to symmetric digraphs $D$4 inside $D$5. Each $D$6 inherits substantial connectivity or is complete symmetric. The case $D$7 (no decomposition) yields direct Hamiltonicity by adaption of Chvátal–Erdős.
3. Gallai–Milgram Path Cover: The exceptional set $D$8 is covered using at most $D$9 directed paths. These are contracted into pseudo-vertices, yielding a reduced digraph $f_2(a)$0.
4. Connectivity Analysis: A contraction lemma shows $f_2(a)$1 inherits high connectivity: $f_2(a)$2.
5. Application of Jackson's Linking Lemma: The final step assembles the symmetric digraphs $f_2(a)$3 (with appropriate orderings) and singleton path vertices into a cover that meets the intricate technical hypotheses of Jackson's lemma, which guarantees a Hamilton cycle in $f_2(a)$4 and thus in $f_2(a)$5 by lifting the contractions.

Careful optimization of the decomposition threshold and combinatorial coefficients leads to the cubic connectivity bound.

Numerical Bounds and Commentary

The theorem supplants Jackson’s exponential bound $f_2(a)$6 with the explicit bound $f_2(a)$7. For small values of $f_2(a)$8, this provides a significant improvement in practical thresholds for forcing Hamiltonicity in digraphs under 2-cycle restrictions. While the Jackson–Ordaz conjecture that $f_2(a)$9 is linear remains unresolved, the new bound makes the distance to this optimal conjecture polynomial rather than factorial.

Implications and Future Directions

This polynomial refinement has several implications:

• Hamiltonicity Theory Progress:

The result delineates the landscape for further reduction toward linearity in Hamiltonicity thresholds in digraphs parameterized by 2-cycle-independent number.

• Techniques for Digraph Decomposition:

The utilization of fixed-threshold decompositions and intricate path covers shows promise in broader applications within digraph theory and related connectivity-Hamiltonicity questions.

• Potential for Algorithmic Development:

Polynomial bounds may imply feasibly computable certificates for Hamiltonicity under strong connectivity and 2-cycle constraints.

Future work could pursue tightening the cubic bound, exploring algorithmic ramifications, and strengthening connections to related conjectures in extremal digraph theory.

Conclusion

The paper establishes a cubic upper bound for the required connectivity to force Hamiltonicity in digraphs with bounded 2-cycle independence number, closing a longstanding significant gap and providing new tools for further refinement in structural and extremal digraph theory (2606.08401).