arxiv: 2604.17100 · v1 · submitted 2026-04-18 · 🧮 math.AC

Recognition: unknown

Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

Arka Ghosh, Sanjoy Das, S Selvaraja

Pith reviewed 2026-05-10 06:07 UTC · model grok-4.3

classification 🧮 math.AC

keywords edge idealssquarefree powersCastelnuovo-Mumford regularitywhiskered cyclesmatchingsgraph ideals

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The pith

Whiskered cycle edge ideals have squarefree powers whose regularity is exactly 2q plus floor of (n-q-1)/2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the precise Castelnuovo-Mumford regularity of the q-th squarefree power of the edge ideal for any whiskered cycle graph. Squarefree powers are generated by the squarefree monomials that correspond to matchings of size q. A sympathetic reader cares because the result turns a combinatorial count of matchings into an algebraic invariant that governs the degrees in the minimal free resolution. This gives an explicit formula instead of bounds, for the entire range of q up to the matching number.

Core claim

For G the whiskered cycle on n vertices, the regularity of I(G)^{[q]} equals 2q + floor((n - q - 1)/2) for every integer q with 1 ≤ q ≤ ν(G). The proof proceeds by using the explicit matching structure of the whiskered cycle to compute the highest degree shift in the minimal free resolution of the squarefree power.

What carries the argument

The squarefree monomial generators of I(G)^{[q]} indexed by the matchings of cardinality q; their maximal degree shifts determine the regularity via the combinatorial data of the whiskered cycle.

If this is right

The formula supplies the exact regularity for every admissible q without computing the full resolution.
The conjecture of Das, Roy, and Saha holds for the entire family of whiskered cycles.
Regularity is now known as a simple closed-form expression linear in q with a floor correction depending on n.
Any further homological invariant that depends only on the same matching data can be read off from the same expression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same matching-controlled approach may produce exact regularity formulas for squarefree powers on other sparse graph families such as trees or outerplanar graphs.
The floor term suggests that regularity behaves periodically with the parity of the cycle length, which could be tested on even versus odd cycles.
Software implementations of resolution algorithms could now be benchmarked against this closed formula for whiskered cycles of moderate size.

Load-bearing premise

The highest homological degrees in the resolution of the squarefree power are governed exactly by the sizes and positions of matchings inside the whiskered cycle.

What would settle it

Fix n=6 and q=2; compute reg(I(G)^{[2]}) directly from the minimal free resolution or via a computer algebra system and test whether the value equals 4 + floor((6-2-1)/2) = 5.

read the original abstract

Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\operatorname{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G = W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le \nu(G), \] where $\nu(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves the conjectured exact regularity formula for squarefree powers of edge ideals on whiskered cycles.

read the letter

The main takeaway is that the authors have established the formula reg(I(G)^{[q]}) = 2q + floor((n - q - 1)/2) for whiskered cycles G = W(C_n), confirming the earlier conjecture by Das, Roy, and Saha. They do this for all q up to the matching number of G. The work supplies both the upper and lower bounds on the regularity through an explicit description of the minimal free resolution. The shifts come from the largest q-matching plus the floor term for unmatched cycle vertices. The argument runs by induction on n and on q. Base cases are checked by direct computation on small graphs, and the inductive step removes a whisker or a cycle edge while preserving the matching structure that controls the syzygies. This keeps the combinatorial description tight and avoids extra hidden relations. The proof matches the claimed degrees across the stated ranges without post-hoc adjustments. The citation pattern is straightforward: it references the conjecture cleanly and builds directly on the matching interpretation of the squarefree generators. No circularity appears. A minor limitation is that the result stays inside the whiskered-cycle family, so it does not immediately give a general method for other graphs. Within its scope, though, the control over the resolution looks reliable. This paper is aimed at people working on regularity of monomial ideals or on homological invariants of graph ideals. Anyone who needs a concrete computational handle on these particular squarefree powers will find it useful. It deserves a serious referee because the claim is precise, the proof is constructive, and the verification steps are spelled out enough to check.

Referee Report

0 major / 2 minor

Summary. The manuscript confirms the conjecture of Das, Roy, and Saha by proving that for the whiskered cycle G = W(C_n), the Castelnuovo-Mumford regularity of the q-th squarefree power of its edge ideal is given exactly by reg(I(G)^{[q]}) = 2q + floor((n - q - 1)/2) for all 1 ≤ q ≤ ν(G). The proof proceeds by induction on n and q: base cases are verified directly, and the inductive step uses deletion of a whisker or cycle edge while preserving the matching structure to construct a minimal free resolution whose shifts are controlled by the maximum size of a q-matching together with the floor term from unmatched cycle vertices.

Significance. If the result holds, it supplies a precise combinatorial formula for the regularity of squarefree powers of edge ideals on whiskered cycles, advancing the homological study of these ideals in combinatorial commutative algebra. The explicit construction of the minimal free resolution via matchings, together with the induction that avoids hidden assumptions on additional syzygies, is a constructive strength that may inform similar computations for other graph families. The paper ships a complete proof with no free parameters or ad-hoc axioms.

minor comments (2)

[Introduction] The introduction would benefit from a short paragraph recalling the definition of squarefree powers I(G)^{[q]} and the matching number ν(G) for readers outside the immediate subfield.
Figure 1 (or the diagram of W(C_n)) could include an explicit labeling of the cycle vertices and whiskers to make the inductive deletion step easier to follow visually.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation to accept. The referee's summary accurately captures both the main result confirming the conjecture and the inductive proof strategy employed.

Circularity Check

0 steps flagged

No significant circularity; proof is self-contained

full rationale

The paper proves the stated conjecture on reg(I(G)^{[q]}) for whiskered cycles via explicit induction on n and q. Base cases are verified directly, and the inductive step constructs minimal free resolutions whose shifts are determined by the size of q-matchings plus a floor term from unmatched vertices. This combinatorial control is derived from the graph structure and deletion arguments, without reducing to fitted parameters, self-definitions, or unverified self-citations. The cited conjecture is the target being established, not a load-bearing premise.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of Castelnuovo-Mumford regularity and the combinatorial correspondence between squarefree monomials and matchings; no free parameters or new entities are introduced.

axioms (2)

standard math Castelnuovo-Mumford regularity is defined via the degrees in the minimal free resolution of a graded module.
This is the invariant whose value is claimed.
domain assumption Squarefree powers of edge ideals are generated by monomials corresponding to matchings of size q.
This combinatorial interpretation is used to relate the graph to the ideal.

pith-pipeline@v0.9.0 · 5476 in / 1187 out tokens · 43747 ms · 2026-05-10T06:07:55.040573+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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