Hamiltonian and Pseudo-Hamiltonian Cycles and Fillings In Simplicial Complexes
Pith reviewed 2026-05-24 19:58 UTC · model grok-4.3
The pith
Hamiltonian d-cycles of complete rank exist in K_n^d for d=2 on certain n and for d=3 on infinitely many n, with near-Hamiltonian simple d-cycles always constructible.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A Hamiltonian d-cycle in K_n^d is a simple d-cycle of complete rank, or equivalently of size 1 + binom(n-1,d). Over F2 and Q, for d=2 such cycles exist precisely when n meets a stated arithmetic condition. For d=3 they exist for infinitely many n. For every d there exist simple d-cycles of size binom(n-1,d) minus O(n^{d-3}). All statements are proved by explicit constructions. The identical claims hold for Hamiltonian d-fillings, which are acyclic fillings of complete rank; removing one simplex from a Hamiltonian cycle yields a Hamiltonian filling of the boundary of that simplex.
What carries the argument
The Hamiltonian d-cycle (or d-filling), defined as a simple acyclic object of complete rank equivalently sized 1 + binom(n-1,d), whose boundary relation lets removal of one simplex produce a filling of the boundary of that simplex.
If this is right
- For d=2 the existence of Hamiltonian 2-cycles is completely characterized by n.
- For d=3 Hamiltonian 3-cycles exist on an infinite arithmetic progression of n.
- Simple d-cycles of size binom(n-1,d) minus O(n^{d-3}) exist constructively for every d and n.
- Hamiltonian d-fillings of any (d-1)-cycle exist under exactly the same conditions as the cycles.
- The cycle-to-filling relation holds over both F2 and Q because the boundary of a cycle minus one simplex equals the boundary of the removed simplex.
Where Pith is reading between the lines
- The explicit constructions make it feasible to generate the objects by computer for moderate n and check further properties such as link connectivity.
- The O(n^{d-3}) gap suggests a natural next question of whether the additive error term can be reduced for each fixed d.
- Because the proofs rely only on linear algebra over F2 and Q, analogous statements may hold over any field in which the rank-size equivalence is valid.
- These objects supply concrete examples of high-dimensional cycles whose homology is trivial except in the top dimension, useful for testing conjectures on the homology of random complexes.
Load-bearing premise
That complete rank is exactly equivalent to having size 1 + binom(n-1,d) over the coefficient fields F2 and Q, and that the boundary operator acts so a cycle minus one simplex fills the boundary of the removed simplex.
What would settle it
An exhaustive search for n=7 or n=8 and d=2 that finds no Hamiltonian 2-cycle on any of the n predicted by the characterization, or a proof that only finitely many n admit Hamiltonian 3-cycles.
read the original abstract
We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} \choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$'s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$'s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} \choose d} - O(n^{d-3})$. All the above results are constructive. Our approach naturally extends to (and in fact, involves) $d$-fillings, generalizing the notion of $T$-joins in graphs. Given a $(d-1)$-cycle $Z^{d-1} \in K_n^d$, ~$F$ is its $d$-filling if $\partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} \choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $\sigma$, then $Z\setminus \sigma$ is a a Hamiltonian $d$-filling of $\partial \sigma$ (a closely related fact is also true for cycles over $Q$). Thus, the two notions are closely related. Most of the above results about Hamiltonian $d$-cycles hold for Hamiltonian $d$-fillings as well.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Hamiltonian d-cycles in the complete simplicial d-complex K_n^d as simple d-cycles of complete rank (equivalently, of size exactly 1 + binom(n-1,d)) over F_2 and Q. It characterizes the n for which such cycles exist when d=2, proves existence for infinitely many n when d=3, and constructs simple d-cycles of size binom(n-1,d) - O(n^{d-3}) in general. Parallel results hold for Hamiltonian d-fillings (acyclic complete-rank fillings of (d-1)-cycles). All results are constructive, and the notions are related by removing one simplex from a cycle to obtain a filling.
Significance. If the rank-size equivalence and boundary-operator properties hold, the constructive existence results generalize Hamiltonian cycles and T-joins to higher dimensions with explicit size bounds, which would be of interest in combinatorial topology.
major comments (2)
- [Abstract / §2 (Definitions)] Abstract and definition of Hamiltonian d-cycle: the asserted equivalence between 'complete rank' and size exactly 1 + binom(n-1,d) (and binom(n-1,d) for fillings) is load-bearing for every existence claim. The manuscript must contain an explicit proof that, over F_2 and Q, the boundary map and rank condition on the support imply this precise cardinality; without it the size-based constructions do not establish the claimed complete-rank objects.
- [d=3 existence / General construction] § on d=3 infinite family and general O(n^{d-3}) construction: the proofs rely on producing objects of the target size and invoking the equivalence; any gap in the rank-size relation would invalidate the Hamiltonian conclusion for these families.
minor comments (2)
- [Abstract] Abstract contains the typo 'a a Hamiltonian'.
- [Throughout] Notation for the size formula uses {{n-1} choose d}; consistent LaTeX rendering and explicit definition of the binomial coefficient in the main text would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for highlighting the need for an explicit justification of the rank-size equivalence. We agree this is essential and will add the required proof in the revision.
read point-by-point responses
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Referee: [Abstract / §2 (Definitions)] Abstract and definition of Hamiltonian d-cycle: the asserted equivalence between 'complete rank' and size exactly 1 + binom(n-1,d) (and binom(n-1,d) for fillings) is load-bearing for every existence claim. The manuscript must contain an explicit proof that, over F_2 and Q, the boundary map and rank condition on the support imply this precise cardinality; without it the size-based constructions do not establish the claimed complete-rank objects.
Authors: We agree that an explicit proof of the equivalence is required. In the revised manuscript we will insert a new lemma (placed in §2 immediately after the definitions) that proves, over both F_2 and Q, that any simple d-cycle (resp. acyclic d-filling) whose support has full rank in the appropriate chain space must have cardinality exactly 1 + binom(n-1,d) (resp. binom(n-1,d)). The argument uses the exact sequence relating the boundary operator on the complete complex K_n^d to the rank of the support and the fact that the only d-chains of full rank are the ones that span the entire homology or the full cycle space. This lemma will be invoked in all subsequent existence statements. revision: yes
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Referee: [d=3 existence / General construction] § on d=3 infinite family and general O(n^{d-3}) construction: the proofs rely on producing objects of the target size and invoking the equivalence; any gap in the rank-size relation would invalidate the Hamiltonian conclusion for these families.
Authors: The constructions in these sections produce objects of the stated cardinalities; once the new lemma establishing the rank-size equivalence is in place, the Hamiltonian (or complete-rank) conclusion follows immediately for the d=3 infinite family and for the general O(n^{d-3}) bound. No further modifications to the constructions themselves are required. revision: yes
Circularity Check
No significant circularity; constructions are independent of the size-rank equivalence
full rationale
The paper states the equivalence between 'complete rank' and size 1 + binom(n-1,d) at the definition stage and then supplies explicit constructions that produce objects meeting the size bound. No step reduces a claimed existence result to a fitted parameter, self-citation chain, or definitional renaming; the boundary-operator facts are standard linear-algebra properties over F2/Q. The derivation chain therefore remains self-contained and does not meet any of the enumerated circularity patterns.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Those are the simple d-cycles of a complete rank, or, equivalently, of size 1 + binom(n-1,d). ... all maximal acyclic sets have the same size, which is binom(n-1,d)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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