Edge-disjoint Hamilton cycles under a bipartite-hole condition

Abstract: In 2017, McDiarmid and Yolov introduced the bipartite-hole-number $\widetildeα(G)$ and proved that $δ(G)\ge \widetildeα(G)$ forces a Hamilton cycle. They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\widetildeα(G)\le a$ and $δ(G)\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=Θ\left(a+k+\frac{ak}{\log(k+2)}\right).$ The upper bound uses a deletion lemma for the bipartite-hole-number together with the McDiarmid--Yolov Hamiltonicity theorem and a greedy packing argument. The lower bound is obtained from three extremal constructions, the logarithmic one using a sparse random auxiliary graph with no prescribed bipartite hole.

• The paper establishes that the optimal minimum degree threshold for k edge-disjoint Hamilton cycles is Θ(a + k + ak/log(k+2)), refining previous bounds.
• It employs a deletion lemma and induction to analyze the impact of edge removals on the bipartite-hole-number, ensuring sufficient connectivity for cycle packing.
• Probabilistic extremal constructions demonstrate the tightness of the new threshold, unifying classical Hamiltonicity conditions with novel bipartite constraints.

Edge-Disjoint Hamilton Cycles Under a Bipartite-Hole Condition

Background and Context

Hamiltonicity conditions in graphs have been a central theme in extremal combinatorics since Dirac’s theorem, which asserts that a graph of order $n \geq 3$ with minimum degree at least $n/2$ is Hamiltonian. Extensions by Ore and Chvátal–Erdős have considered more intricate structural constraints. More recently, McDiarmid and Yolov introduced the bipartite-hole-number $\widetilde{\alpha}(G)$, defined as the minimal integer $r$ such that for some $s, t$ with $r = s + t - 1$, $G$ contains no $(s, t)$-bipartite hole, i.e., no pair of disjoint sets $S, T$ of sizes $s, t$ respectively with no edges between $n/2$0 and $n/2$1. This parameter interpolates between independence and density-type Hamiltonicity conditions.

McDiarmid and Yolov's result guarantees Hamiltonicity if $n/2$2. They also provided a sufficient degree condition for packing $n/2$3 edge-disjoint Hamilton cycles, specifically $n/2$4. However, the optimality of this threshold for $n/2$5 remained unresolved, motivating the present work.

Main Results

The core contribution is the determination of the asymptotically optimal minimum degree threshold $n/2$6, for the existence of $n/2$7 edge-disjoint Hamilton cycles in a graph $n/2$8 satisfying $n/2$9, up to absolute constant factors. Explicitly, the authors prove:

$\widetilde{\alpha}(G)$0

This removes the necessity for the $\widetilde{\alpha}(G)$1 mixed term in the earlier sufficient condition, replacing it by a strictly smaller $\widetilde{\alpha}(G)$2 term, which is shown to be tight up to constant factors.

The proof is separated into two major components:

• Upper Bound: Employing a deletion lemma for the bipartite-hole-number and an explicit induction on packing Hamilton cycles, they show that a minimum degree of $\widetilde{\alpha}(G)$3 suffices, for some absolute constant $\widetilde{\alpha}(G)$4.
• Lower Bound: Three independent extremal constructions demonstrate necessary lower bounds for the $\widetilde{\alpha}(G)$5, $\widetilde{\alpha}(G)$6 and $\widetilde{\alpha}(G)$7 terms, specifically leveraging random graph techniques to exhibit the necessity of the logarithmic denominator in the mixed term.

Technical Approach

The upper bound fundamentally improves the quantitative profile of packing constraints. The central technical innovation is a detailed analysis of how the bipartite-hole-number behaves under edge deletions with bounded maximum degree. The deletion lemma quantifies the increase in $\widetilde{\alpha}(G)$8 after removing a sparse subgraph, ensuring sufficient connectivity remains to iteratively find Hamilton cycles.

To operationalize this, the authors introduce the function $\widetilde{\alpha}(G)$9, providing upper bounds on the bipartite-hole-number after deleting a subgraph of maximum degree $r$0. They then show that if for $r$1,

$r$2

then $r$3 edge-disjoint Hamilton cycles can be greedily constructed.

The lower bound constructions, particularly for the $r$4 term, utilize probabilistic arguments. A bipartite random auxiliary graph is engineered to have no $r$5-bipartite hole and with constrained edge density, such that any $r$6 edge-disjoint Hamilton cycles in the full construction would require more edges than exist, leading to the required threshold.

Implications and Theoretical Advances

This result sharply characterizes the interplay between structural density (via the minimum degree), forbidden substructures (through the bipartite-hole-number), and edge-disjoint Hamiltonicity. Namely, it confirms that the combinatorial obstruction to packing many Hamilton cycles, given only these parameters, is strictly less severe than previously thought—by a factor of $r$7 in the mixed term.

Several notable implications arise:

• Tightness: The logarithmic savings cannot be eliminated within this parameter regime; the random construction in the lower bound removes the possibility of improvement beyond constants.
• Extremality: The constructions highlight precisely when and why more intricate graph structure (beyond degree and bipartite-hole constraints) is needed, especially in extremal and random graphs.
• Interplay with Classical Results: This work generalizes and subsumes various classic minimum degree and independence number-based Hamiltonicity and packing results, showing their special cases.

Numerical Strength and Strong Claims

The explicit bounds are of the form:

$r$8

for universal constants $r$9 and all integers $s, t$0. The improvement over previous sufficient conditions is substantial in regimes where $s, t$1 are large, with the $s, t$2 correct up to constants and the necessity demonstrated by explicit constructions.

Future Directions

This work suggests several further questions:

• The determination of optimal absolute constants in front of each asymptotic term, especially $s, t$3, which may be of interest both for theory and applications.
• Whether finer-grained properties or additional graph invariants refine the feasibility threshold for edge-disjoint Hamilton cycle packing.
• Potential algorithmic applications: These structural insights may inform efficient algorithms for packing Hamilton cycles in large sparse graphs.

Conclusion

The order-of-magnitude minimum degree threshold for packing $s, t$4 edge-disjoint Hamilton cycles in graphs constrained by the bipartite-hole number is established as $s, t$5, up to universal constants. This resolves an open problem and sharply quantifies the joint effect of sparsity and independence-like obstructions on Hamiltonicity in edge-disjoint settings. The methods unify probabilistic, extremal, and iterative techniques, and the framework potentially guides further refinement in Hamiltonicity theory.