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Gracefulness of two nested cycles: a first approach

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Pith's one-line read Two nested cycles admit graceful labelings in every congruence class that parity allows, once the outer cycle is sufficiently long.

desk verdict Plausible new constructions for graceful two nested cycles, but the snowflake classification is not proven as stated because Lemma 4.16 fails for stars of sizes 1 and 2. read the letter →

arxiv 2411.12998 v1 pith:Y4ZRACKO submitted 2024-11-20 math.CO

Miguel Licona , Joaquín Tey This is my paper · ORCID
classification math.CO MSC 05C7805C0505C21
keywords gracefulgraphnear-gracefulconservativetreelabelingsemidualtwonestedcyclessnowflakeSkolemsystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the semidual of a plane graph made of two nested cycles is a snowflake, and that every snowflake with $M$ edges is conservative when $M \equiv 0,3 \pmod 4$ and near-conservative otherwise. It then builds graceful labelings directly: for every inner cycle length $m_1 \ge 3$ and every outer cycle length $m_2 \ge m_1(2m_1-1)$, a graceful two nested cycles graph exists when $m_1+m_2 \equiv 0,3 \pmod 4$, and a near-graceful one exists when $m_1+m_2 \equiv 1,2 \pmod 4$. This matters because a long-known parity obstruction forbids graceful labelings of graphs whose vertices all have even degree when the edge count is $1$ or $2$ mod $4$, and the paper shows that, for long outer cycles, that obstruction is the only obstacle. The proof works by translating graceful vertex labelings into zero-sum edge labelings of the semidual, where the problem reduces to signing numbers on the branches of a star tree.

What carries the argument

The central object is the snowflake, the tree obtained from a galaxy of stars by identifying one leaf from each star at a single vertex; this is exactly the semidual of a two nested cycles graph. The labeling machinery is built from $t$-Skolem sequences, partitions of $\{1,2,\dots,2n-1\}\cup\{2n+t\}$ into $n$ pairs whose differences are $1,2,\dots,n$, repackaged as zero-sum Skolem systems that assign signed edge labels to each star so that every internal vertex has zero sum. Attachment lemmas (Proposition 3.6, Lemmas 4.6--4.15) show how to glue a conservative or near-conservative snowflake to stars that admit balanced (Eulerian) conservative labelings, by matching vertex-sums at the joining vertex. The proof of Theorem 4.17 splits snowflakes into even snowflakes, handled by parity cases, and non-even snowflakes reduced to a minimum snowflake with star sizes $3,4,5,6$ plus extra stars of sizes divisible by $4$; the reduction is Lemma 4.16.

What would settle it

A concrete test: exhaustively search all snowflakes with up to, say, 20 edges and check whether every one with $M \equiv 0,3 \pmod 4$ admits a zero-sum labeling with zero vertex-sum at each internal vertex; the first failure would refute Theorem 4.17, while a full pass would isolate any remaining gap to the missing decomposition proof.

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Extended reading notes

Core claim

On its own terms, the paper establishes a complete congruence classification for snowflakes: a snowflake of size $M$ is conservative exactly when $M \equiv 0,3 \pmod 4$, and near-conservative when $M \equiv 1,2 \pmod 4$ (Theorem 4.17). Since the semidual of a two nested cycles graph is a snowflake, this characterizes when that semidual is conservative or near-conservative. The second result is an existence theorem for the original graphs: for any $m_1 \ge 3$, whenever $m_2 \ge m_1(2m_1-1)$ and $m_1+m_2 \equiv 0,3 \pmod 4$, the set $N_g(m_1,m_2)$ of graceful two nested cycles graphs is nonempty, and when $m_1+m_2 \equiv 1,2 \pmod 4$ the set $N_{n-g}(m_1,m_2)$ of near-graceful ones is nonempty (Theorem 2.1).

Load-bearing premise

The proof depends on the claim, stated without a detailed demonstration, that every non-even snowflake can be obtained by gluing stars whose sizes are multiples of four onto a small 'minimum' snowflake, and Theorem 2.1's four-case construction is only written out for one case, with the other three described as similar.

Editorial extensions

If this is right

  • Every snowflake with $M \equiv 0,3 \pmod 4$ edges admits a conservative labeling, and every other snowflake admits a near-conservative labeling, so this tree family is fully classified.
  • For each $m_1 \ge 3$ and $m_2 \ge m_1(2m_1-1)$, the congruence of $m_1+m_2$ alone determines whether a graceful or near-graceful two nested cycles graph exists.
  • The known parity obstruction for graphs whose vertices all have even degree is sharp in this family: in the excluded residue classes the near-graceful labeling is the best possible outcome.
  • The semidual viewpoint reduces a problem about vertex labelings of even-degree plane graphs to a problem about signed edge labelings of star trees, giving a template for studying other plane cycles with chords.
  • The paper leaves open whether every two nested cycles graph with an allowed total size is graceful, not just those with a sufficiently long outer cycle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same star-attachment machinery, if the decomposition lemma is proved in full, would likely extend the conservative and near-conservative classification to other trees whose internal degrees are controlled, not just snowflakes.
  • The threshold $m_1(2m_1-1)$ is probably not the true minimum; the omitted 'similar' cases suggest the construction may work for much smaller $m_2$, and small-case computation could reveal the actual bound.
  • Because conservative snowflake labelings correspond to zero-sum families of signed integer sequences (Skolem systems), the construction doubles as an existence proof for design-theoretic objects such as cyclic cycle decompositions or Heffter arrays with prescribed branch sizes.
  • A computational search over small two nested cycles graphs could turn the paper's closing question into a conjecture: check whether every graph with $M \equiv 0,3 \pmod 4$ is graceful, using Theorem 4.17 to filter semiduals that admit conservative labelings first.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper addresses two related graph-labeling questions. Theorem 2.1 asserts that for every m1 ≥ 3 and every m2 ≥ m1(2m1−1), there is a two nested cycles plane graph that is graceful if m1+m2 ≡ 0 or 3 (mod 4) and near-graceful if m1+m2 ≡ 1 or 2 (mod 4). The second main result, Theorem 4.17, asserts that every snowflake (a tree obtained by identifying one leaf from each of a disjoint family of stars) of size M is conservative when M ≡ 0 or 3 (mod 4) and near-conservative otherwise. The proofs are constructive and rely on Skolem systems and attachment lemmas from the literature, connecting conservativeness of snowflakes with gracefulness of two nested cycles via the semidual construction.

Significance. If completed, the results would provide a clean semidual obstruction and a new infinite family of graceful Eulerian plane graphs, which is notable because most previously studied graceful cycles with chords are non-Eulerian. The paper is explicitly a first approach and has the virtue of giving explicit labelings in the written cases. The reliance on [13] and [16] is not circular: the new statements are existence results with explicit constructions. However, the manuscript as submitted does not establish the full theorems, because several load-bearing case analyses are omitted and one decomposition lemma is false on its stated domain.

major comments (3)
  1. [Section 2, proof of Theorem 2.1] Only Case 1 of the four-case construction is proved. Cases 2, 3, and 4 are dismissed with the statement that the analysis is similar and omitted; but the theorem's near-graceful conclusions for m ≡ 1, 2 (mod 4) rest on Cases 2 and 4, and Case 3 covers a parity subcase of the graceful conclusion. The examples in Figures 2–5 illustrate only m1 = 3 and m1 = 4 and do not constitute a proof for all parameters. The omitted cases must be written out or replaced by a uniform argument that covers all four parity combinations.
  2. [Section 4, Lemma 4.16] The decomposition used in the proof of Lemma 4.16 is not valid for all snowflakes allowed by Definition 4.1. Definition 4.1 permits stars of any positive size, while a minimum snowflake uses only sizes 3, 4, 5, 6 and the attached stars are required to have sizes divisible by 4. A snowflake containing a star of size 1 or 2 cannot be represented in this form: for instance C_{2,3} has size 5, is non-even (it has an internal vertex of degree 3), and admits no decomposition into a minimum snowflake plus stars of sizes divisible by 4. Since Lemma 4.16 is the entire input to Theorem 4.17, the classification of snowflakes is not proved for the full class stated. The authors should either restrict Definition 4.1 and Theorem 4.17 to exclude sizes 1 and 2, or provide an additional argument covering those cases.
  3. [Section 4, Lemmas 4.7, 4.8, 4.10, 4.12, 4.15] Several lemmas used in the proof of Theorem 4.17 omit essential subcases. Lemma 4.7 Case 2, Lemma 4.8 Case 2, Lemma 4.10 Case 3.2, and Lemma 4.15 Case 1.2 are all declared analogous and not proved; Lemma 4.12 is dismissed as easy. These omissions are load-bearing: Lemmas 4.7 and 4.8 together prove Theorem 4.9 for even snowflakes, and Lemma 4.15 is needed for the non-odd minimum snowflakes in Lemma 4.16. A revised manuscript should either supply the omitted arguments or clearly reduce them to the proved cases with explicit justification.
minor comments (4)
  1. [Section 4, Lemma 4.16] In the sentence describing the attachment, 'aforth' should be 'aforementioned' and 'most be identified' should be 'must be identified'.
  2. [Section 4, Lemma 4.14] In the bullet points, the expressions 'M + k − t + 2' appear where 'M + k + t + 2' seems intended; please check the consistency of the signs.
  3. [Section 4, Lemma 4.15] The notation 'sC2(z2)' lacks the arrow over C2 and should be written as 's_{\vec{C}_2}(z_2)' as in the rest of the paper.
  4. [Section 2, proof of Theorem 2.1] The intervals defining f(e_i) may appear empty for small values of m2; a sentence confirming that the hypothesis m2 ≥ m1(2m1−1) guarantees all intervals are nonempty in all four cases would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main theorems are derived from explicit labelings and independent prior lemmas; the unproved decomposition in Lemma 4.16 is a correctness gap, not a circular reduction.

full rationale

The derivation chain is not circular. Theorem 4.17 is built from independent prior results (stars and galaxies are conservative via [13], and the attachment Proposition 3.6 from [16]) together with new Skolem-system lemmas 4.10–4.15 that construct labelings of minimum odd snowflakes. None of these cited lemmas states or assumes that every snowflake, or the semidual of a two-nested-cycles graph, is conservative, so the target theorem is not an input to its own proof. Theorem 2.1 is also an explicit construction: the edge-labeling f and vertex-labeling phi are written out, and the verification in Case 1 is a direct set-equality; the other cases are merely omitted as “similar,” which is an incompleteness, not circularity. The main substantive concern is Lemma 4.16, which asserts without proof that every non-even snowflake decomposes into a minimum snowflake plus stars whose sizes are multiples of four. That assertion is not justified by Definition 4.1 and appears not to cover stars of size 1 or 2, so Theorem 4.17 as stated may not be fully established for all snowflakes. However, this is a domain/completeness gap, not a circular step: the claimed decomposition, if true, would feed independent lemmas, and if false, the theorem is unproved rather than equivalent to its assumptions. The self-citations [13] and [16] are load-bearing, but they are prior published results whose statements do not include the present theorem, so they do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or fitted values appear; the paper is a pure existence proof. The axioms are standard theorems plus two undischarged proof obligations: the omitted case arguments and the snowflake decomposition lemma.

assumptions (7)
  • standard math Rosa's theorem: an Eulerian graph with |E(G)| ≡ 1,2 (mod 4) has no graceful labeling.
    Used in Section 1 to justify that only near-graceful labelings are possible for total edge count 1 or 2 modulo 4.
  • standard math The dual/semidual transfer: a graceful plane graph has a conservative dual and a near-graceful plane graph has a near-conservative dual (Theorem 1.2 and its adapted version).
    Bridges the graceful labeling problem to the conservative snowflake problem in Sections 1 and 4.
  • standard math Existence of t-Skolem sequences (Theorem 3.7, from Skolem and O'Keefe).
    Provides the integer partition structure used to construct zero-sum Skolem systems encoding conservative labelings.
  • standard math The (3·n; k; t)-Skolem system with R-order exists when n ≡ t+1,-t (mod 4) (Lemma 3.12 from [13]).
    Cited from the authors' earlier paper; used in Lemma 4.10 to convert Skolem systems into zero-sum labelings.
  • standard math The mixed (3·n1, 5·n2; k; t)-Skolem system exists under the stated congruence conditions (Lemma 3.17 from [13]).
    Cited from prior work; used to label snowflakes built from 3-stars and 5-stars in Section 4.
  • ad hoc to paper The omitted 'similar' cases in Theorem 2.1 and in Lemmas 4.7, 4.8, 4.10, 4.15, and 4.12 actually follow by analogy.
    The paper explicitly declines to write these derivations, so the theorems depend on an unverified symmetry of the constructions.
  • ad hoc to paper Lemma 4.16's decomposition of every non-even snowflake into a minimum snowflake plus stars of size divisible by four.
    Stated without proof and load-bearing for Theorem 4.17.

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Pith. "Pith review of Gracefulness of two nested cycles: a first approach." pith.science (2026). https://pith.science/paper/Y4ZRACKO

@misc{pith2026241112998,
  author       = {Pith},
  title        = {Pith review of: Gracefulness of two nested cycles: a first approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4ZRACKO}},
  note         = {Machine review of arXiv:2411.12998}
}
abstract

It is known that if a plane graph is graceful (resp. near-graceful), then its semidual is conservative (resp. near-conservative). In this work we prove that the semidual of a plane graph of size $M$ consisting of two nested cycles is conservative if $M \equiv 0,3 \pmod 4$, and near-conservative otherwise. We also show that for a given integer $m_1 \geq 3$, there exists $m^* > m_1$ such that for $m_2 \geq m^*$, if $m_1+m_2 \equiv 0,3 \pmod 4$ (resp. $m_1+m_2 \equiv 1,2 \pmod 4$), then there exists a graceful (resp. near-graceful) plane graph consisting of two nested cycles with sizes $m_1$ and $m_2$, respectively.

Figures

Figures reproduced from arXiv: 2411.12998 by the authors.

Figure 1
Figure 1. A two nested cycles graph (in black) and its semidual (in gray dashed lines). [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Example of Case 1. Ng(3, 16) ̸= ∅ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Example of Case 2. Nn-g(3, 18) ̸= ∅. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Example of Case 3. Ng(3, 17) ̸= ∅ [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Example of Case 4. Nn-g(3, 19) ̸= ∅. Now, we will show examples when m1 ≡ 0 (mod 2), more specific when m1 = 4, and so cw = 6. A number with an upper-bar in the given sequence (ϕ(v1), . . . , ϕ(vm2 )) represents a (labeled) vertex of Cm1 . Example of Case 1. Let m2 = 2…
Figure 6
Figure 6. Figure 6: a) An Eulerian conservative labeling. b) An Eulerian near-conservative labeling. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: a) A conservative labeling. b) A near-conservative labeling. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Zero-sum Skolem systems with n = 5 (left) and n = 9 (right) (viewed as snowflakes). Let s ≥ 3. Notice that, by Remark 3.13, D2s+3 = {(−4, −(n + 6s + k), n + 6s + 4 + k}. Moreover, for all 1 ≤ i ≤ 2s + 2 and 2s + 4 ≤ i ≤ 4s + 1 there exist di ∈ abs(Di) and di+1 ∈ abs(Di…
Figure 9
Figure 9. Figure 9: A zero-sum (3 · 6; k; 1)-Skolem system (viewed as a snowflake). For s ≥ 2, we consider two cases. Case 3.1. Suppose s is even. By Lemma 3.15, for some q odd, there exist d ∈ abs(Dq) and d ′ ∈ abs(Dq+1) such that |d − d ′ | = 2. In fact, for q = 3s + 1 take d = n + 7s +…
Figure 10
Figure 10. Figure 10: Zero (3 · 3, 5 · 1, ; k; 1)-Skolem system. 4.3. Attaching minimum odd snowflakes with even snowflakes Lemma 4.12. Let x ∈ N, ℓ ∈ {x − 1, x, x + 1}, C an even snowflake of size M such that its center z has odd degree n ≥ 3 and {vi} n 1 = VI (C) \ {z}. Then there exist …

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