Planar Tur\'an number of two adjacent cycles

Guiying Yan; Qiang Zhou; Xinzhe Song

REVIEW 3 major objections 4 minor 15 references

Planar Tur\'an number of two adjacent cycles

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The planar Turán numbers of two small adjacent cycles are now known exactly: $C_3\text{-}C_3$ gives $3n-6$ for $n\le5$, $3n-7$ for $n=6$, and $\lceil5n/2\rceil-5$ for $n\ge7$; $C_3\text{-}C_4$ gives $3n-6$ for $n\le6$, $3n-7$ for $n=7$…

desk verdict Theorem 1.1 for C3-C3 looks like a real new result, but Theorem 1.2 for C3-C4 is not proved as written: Lemma 4.3's t=0 case has a genuine logical gap, and Corollary 1.2 contradicts the paper's own construction and Li's result. read the letter →

arxiv 2411.18487 v2 pith:BJ3DGIRN submitted 2024-11-27 math.CO

Xinzhe Song , Guiying Yan , Qiang Zhou This is my paper · ORCID

classification math.CO MSC 05C3505C10

keywords planarTuránnumberadjacentcyclesC3-C3C3-C43-face-blockdischargingmethodextremalgraphsdisjoint

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

This paper determines the exact planar Turán number for two subgraphs that each consist of a pair of small cycles joined by a single edge: two triangles ($C_3\text{-}C_3$) and a triangle joined to a 4-cycle ($C_3\text{-}C_4$). For $C_3\text{-}C_3$, the maximum number of edges in an $n$-vertex planar graph avoiding it is $3n-6$ for $n\le 5$, $3n-7$ for $n=6$, and $\lceil 5n/2\rceil-5$ for $n\ge 7$. For $C_3\text{-}C_4$, the corresponding values are $3n-6$ for $n\le 6$, $3n-7$ for $n=7$, and $\lfloor 5n/2\rfloor-4$ for $n\ge 8$. The proof partitions the vertex set into maximal '3-face-blocks' and shows that the total number of triangular faces in an extremal graph is bounded by $n-1$ (for $C_3\text{-}C_3$) or $n$ (for $C_3\text{-}C_4$), a bound that then forces the stated edge counts via Euler's formula. A by-product is a correction of a previously published value for the disjoint-union case $2C_3$ at $n=6$.

What carries the argument

The load-bearing object is the $3$-face-block $B$, a maximal connected set of vertices in which every edge of every connecting path lies on some triangular face. For a plane graph $G$, each vertex $v$ has a count $|R_v|$ of triangular faces incident with $v$, and $\sum_{v} |R_v| = 3f_3(G)$. The proofs of Theorem 1.1 and 1.2 hinge on Lemmas 3.3 and 4.3, which classify the $3$-face-blocks for which $\sum_{v\in B}|R_v|$ can be as large as $3|V(B)|$ or $3|V(B)|+3$; outside these exceptional blocks the sum is at most $3|V(B)|-3$ in the $C_3\text{-}C_3$ case and at most $3|V(B)|$ in the $C_3\text{-}C_4$ case. This classification is what lets the authors transfer the forbidden-subgraph condition into a global bound on the number of triangular faces.

What would settle it

Search for a $C_3\text{-}C_3$-free plane graph on $n\ge7$ vertices with more than $\lceil 5n/2\rceil-5$ edges, or find a 3-face-block not isomorphic to $B_1,B_2,B_3$ with $\sum_{v\in B}|R_v|>3|V(B)|$; either would refute Theorem 1.1. For Theorem 1.2, a $C_3\text{-}C_4$-free plane graph on $n\ge8$ vertices with more than $\lfloor 5n/2\rfloor-4$ edges would refute it. A computer search over plane graphs with $n\le10$ can check whether the edge bounds are already tight at small sizes.

Watch

Extended reading notes

Core claim

The central claim is that the planar Turán numbers of $C_3\text{-}C_3$ and $C_3\text{-}C_4$ are exactly as given in Theorems 1.1 and 1.2: the formulas are piecewise linear, matching the maximal planar value $3n-6$ only for $n\le5$ and $n\le6$ respectively, then dipping by roughly $n/2$. The upper-bound proof is built on a discharging scheme over $3$-face-blocks; the key lemma classifies the only blocks in which the sum of incident $3$-face counts can exceed three times the number of vertices, and shows those exceptions cannot occur in a $C_3\text{-}C_3$-free or $C_3\text{-}C_4$-free planar graph without creating the forbidden subgraph. From the classification, the authors deduce $f_3(G)\le n-1$ in the $C_3\text{-}C_3$-free case and $f_3(G)\le n$ in the $C_3\text{-}C_4$-free case, and Euler's formula converts these into the stated edge bounds. The extremal constructions are explicit: for $C_3\text{-}C_3$, a path plus two adjacent apex vertices, one joined to a maximum independent set containing the path ends; for $C_3\text{-}C_4$, a partition $(4,1,1,\ldots)$ fan-like graph.

Load-bearing premise

The argument depends on the classification of exceptional 3-face-blocks in Lemmas 3.3 and 4.3: if any unlisted block has a larger sum of incident triangular-face counts than allowed, the bounds $f_3\le n-1$ (or $n$) and hence the edge counts could fail.

Editorial extensions

If this is right

The exact value of $\operatorname{ex}_{\mathcal P}(n,2C_3)$ is now known for all $n\ge 3$, with the previous small-$n$ error corrected: $3n-6$ for $n\le5$, $3n-7$ for $n=6$, $\lceil 5n/2\rceil-5$ for $n\ge7$ (Corollary 1.1).
The planar Turán number of $C_3\cup C_4$ is determined for every $n\ge3$ (Corollary 1.2), removing the earlier restriction $n\ge20$.
The upper-bound proofs imply $f_3(G)\le n-1$ for connected planar $C_3\text{-}C_3$-free graphs and $f_3(G)\le n$ for connected planar $C_3\text{-}C_4$-free graphs, which is the engine behind the edge bounds.
Extremal constructions are given explicitly for both cases, achieving the stated edge counts for all $n$ at and above the thresholds.

Reading between the lines

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

If the classification lemmas hold, the same technique likely applies to other adjacent-cycle pairs, such as $C_4\text{-}C_4$ or $C_3\text{-}C_5$, though the exceptional-block case analysis would grow substantially; the resulting values may again be piecewise linear with slope $5/2$.
The correctness of the small-$n$ correction for $2C_3$ suggests that other published planar Turán numbers for small $n$ deserve re-checking; the paper's own $n=6$ and $n=7$ exceptional cases show that thresholds can occur beyond what asymptotic constructions suggest.
One could test the classification computationally: exhaustively enumerate all connected plane graphs up to moderate $n$ and look for $3$-face-blocks not among the listed exceptions but with surplus $\sum_{v\in B}|R_v|>3|V(B)|$; if any exist, the corresponding forbidden subgraph must also appear, which would provide a check on Lemmas 3.3 and 4.3.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: Theorem 1.1 for C3-C3 looks like a real new result; Theorem 1.2 for C3-C4 is plausible but not proved as written. Two concrete problems: Lemma 4.3 Case 4.5 contains a non sequitur, and Corollary 1.2 is internally inconsistent.

The genuinely new material is the exact value of ex_P(n,C3-C3) with explicit extremal graphs, plus the small correction to ex_P(6,2C3) (10 to 11). The lower-bound constructions are concrete and checkable. The discharging framework for the C3-C3 upper bound is coherent, and the special blocks B1-B3 are drawn and counted explicitly. The four bad blocks for C3-C4 in Figure 4 are also clearly identified, and the reduction of the upper bound to Lemma 4.3 is easy to follow.

The main soft spot is Case 4.5. To handle t=0 (partition (1,1,...,1)), the text derives a global inequality 3f_3(G)=e_3 ≤ e and then asserts the block-level bound sum_{u in R_v}|R_u| ≤ 3|V(R_v)|. That does not follow. The global edge count e includes edges outside the block; a block could have surplus if other parts of the graph have compensating slack. Since Lemma 4.3's classification is the load-bearing premise for Theorem 1.2, this gap has to be fixed before the C3-C4 upper bound can be accepted.

Second, Corollary 1.2 states ex_P(n,C3∪C4) = floor(5n/2)-5 for n≥8, but the introduction reports Li's result floor(5n/2)-4 for n≥20, and the paper's own Figure 7 construction has floor(5n/2)-4 edges and is claimed to be C3∪C4-free. So the corollary is wrong as written. It is probably a typo, but an internal contradiction of this kind matters in a paper that also needs close checking.

Third, both Lemma 3.3 and Lemma 4.3 rely on repeated 'by a similar argument' and 'easy to verify' steps. Some are probably fine, but the text does not allow an independent reader to verify them without reconstructing each subcase from scratch. A referee will need to go through them by hand.

Bottom line: this paper is worth a serious referee, but as-is it should not be the final version. The C3-C3 result may well stand. The C3-C4 upper bound needs a repaired Lemma 4.3, and Corollary 1.2 needs correction. I would not cite it yet; after a clean revision I'd cite the C3-C3 theorem.

Referee Report

3 major / 4 minor

Summary. The paper determines the planar Turán numbers ex_P(n, C3-C3) and ex_P(n, C3-C4) for all n ≥ 3, with formulas 3n−6 for small n, 3n−7 at the exceptional small value, and approximately 5n/2 for larger n. The upper bounds are obtained by partitioning the vertex set into 3-face-blocks and proving surplus bounds via lengthy case analyses (Lemmas 3.3 and 4.3), then applying Euler's formula and face-counting inequalities. The lower bounds are given by explicit constructions, one based on a path with an added adjacent pair and one generated from a partition (4, 1, 1, ...). The paper also derives corollaries for ex_P(n, 2C3) and ex_P(n, C3 ∪ C4) and claims a correction to a result of Lan, Shi, and Song.

Significance. If the proofs are completed, these would be exact planar Turán numbers for two natural 'adjacent cycles' configurations, going beyond the known disjoint-union cases. The lower-bound constructions are explicit and the claimed edge counts match the stated formulas, including the half-integral values. The 3-face-block discharging strategy is coherent and potentially reusable. However, the central classification lemmas contain many unchecked case arguments, one case in Lemma 4.3 appears not to prove the required local bound, and Corollary 1.2 is internally inconsistent with both the quoted theorem of Li and the paper's own lower-bound construction. The main theorems may well be true, but the manuscript as written does not yet give a complete proof.

major comments (3)

[Section 4, Lemma 4.3, Case 4.5] The treatment of the case t = 0 does not establish the block-surplus bound. After choosing a minimal counterexample, the text derives e_{3,3} = 0 and then '3f_3(G) = e_3 ≤ e', and immediately concludes 'for t = 0, we have Σ_{u∈R_v} |R_u| ≤ 3|V(R_v)|'. This does not follow: e_3 ≤ e is a global inequality (Property 2.1) bounding the total number of 3-faces in the whole graph, and it is compatible with one block having Σ|R_u| > 3|V(B)| when other blocks have compensating slack. What is needed is the local bound f_3(B) ≤ |V(B)| for the block under consideration; the text neither proves that a minimal counterexample can be taken to be the whole graph nor shows how the global slack localizes. Since the proof of Theorem 1.2 reduces the upper bound to Lemma 4.3, this gap is load-bearing.
[Introduction and Corollary 1.2] The claimed deduction of ex_P(n, C3 ∪ C4) is internally inconsistent. The introduction states that Li [12] proved ex_P(n, C3 ∪ C4) = ⌊5n/2⌋ − 4 for n ≥ 20, and the paragraph after Theorem 1.2 asserts that the authors' extremal graphs are also C3 ∪ C4-free. The construction after Theorem 1.2 satisfies v = 2t + 6 and e = 5t + 11 = ⌊5n/2⌋ − 4 for t ≥ 1, i.e., for all n ≥ 8. Thus, if the asserted C3 ∪ C4-freeness is correct, the lower bound for C3 ∪ C4 is ⌊5n/2⌋ − 4, not the ⌊5n/2⌋ − 5 stated in Corollary 1.2; if the construction is not actually C3 ∪ C4-free, then the corollary is unsupported. Either way, the manuscript needs to reconcile this contradiction.
[Lemmas 3.3 and 4.3] The exceptional-block classification is the load-bearing premise of both upper-bound proofs, but many of its steps are asserted rather than proved. Examples include 'by an argument similar to that above' in Case 3.2, 'by a similar discussion' in Cases 3.3 and 3.4, and repeated uses of 'we could request that' in Lemma 4.3. In Lemma 4.3, Case 4.3, after positing |R_{v1}| ≥ 2 and |R_{v4}| ≥ 2, the statement 'no more 3-face can be incident with v8, a contradiction' is not justified by the preceding text. Since an unlisted exceptional block with positive surplus would invalidate the face-counting step f_3(G) ≤ n in the proof of Theorem 1.2, these case analyses need to be written out in full or replaced by a verifiable finite computation.

minor comments (4)

[Section 4, Lemma 4.3, Figure 4] The four special blocks are presented only pictorially; explicitly listing their vertex counts, edge counts, and face counts would make the subsequent inequalities in the proof of Theorem 1.2 much easier to verify.
[Section 4, proof of Theorem 1.2] In the case m < k1 + k2, the bound e(G3) + e(G2 : G3) ≤ 12k1 + 14k2 + 3(m + k1 + k2)/2 − 3 implicitly assumes that the blocks B2 and B3 each have 6 vertices and 12 edges and that B4 has 7 vertices and 14 edges; these counts should be stated explicitly.
[Section 4, Lemma 4.3] The proof repeatedly concludes 'Σ_{u∈R_v}|R_u| ≤ 3|V(R_v)|' and then uses this as a bound for the whole block B; the relationship between V(R_v) and V(B) should be spelled out whenever this identification is used.
[Throughout] There are numerous typos and broken sentences, including 'Moerover', 'conncect', '3-face-bolck', and 'v‘' in the k1 = 3 paragraph of Case 4.4; these should be cleaned up in the revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the extremal bounds are derived from face-counting and explicit constructions, with no fitted parameters or self-referential definitions.

full rationale

The paper's derivation chain is self-contained. The upper bounds for both theorems are obtained by classifying 3-face-blocks (Lemmas 3.3 and 4.3) and then applying the counting identity 3f3(G) = sum_v |R_v| together with Euler's formula; the lower bounds come from explicit plane graph constructions. No parameter is fitted to the extremal value, and no theorem is assumed that is equivalent to the target result. The citations to earlier work on ex_P(n,2C3) and ex_P(n,C3∪C4) are used only for comparison and are not load-bearing for the main proofs. The skeptical concern about Case 4.5 of Lemma 4.3—where the global inequality e_3 <= e is invoked to conclude a per-block surplus bound—is a potential correctness gap in a case analysis, not a circular reduction: the desired block inequality is not defined as the input, and the later counting argument could fail only if that case analysis is incomplete. That is a proof-validity issue, not self-referential derivation, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof introduces no fitted parameters or new postulates. It relies on standard planar graph identities and on the 3-face-block decomposition, which is a definition rather than an empirical entity.

assumptions (3)

standard math Euler's formula for connected plane graphs: n - e + f = 2
Used in the final counting steps of both theorem proofs.
standard math Face-edge identity 2e(G) = sum_{i>=3} i f_i(G)
Property used in Sections 3 and 4 to relate edge count to face counts.
domain assumption Each edge of a 3-face-block lies in at least one triangular face and blocks partition V(G)
Definition 2.1 and Remark 2.1; the partition property is asserted without proof but follows from the definition of maximal connected components.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Planar Tur\'an number of two adjacent cycles." pith.science (2026). https://pith.science/paper/BJ3DGIRN

@misc{pith2026241118487,
  author       = {Pith},
  title        = {Pith review of: Planar Tur\'an number of two adjacent cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJ3DGIRN}},
  note         = {Machine review of arXiv:2411.18487}
}

abstract

The planar Tur\'an number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Tur\'an number of $k(k\geq 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Tur\'an number of $C_{3}\text{-}C_{3}$ and $C_{3}\text{-}C_{4}$, where $C_{k}\text{-}C_{\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them.

Figures

Figures reproduced from arXiv: 2411.18487 by the authors.

**Figure 2.** The 3-face-blocks with P v∈B |Rv| ≥ 3|V (B)| Lemma 3.3. For each 3-face-block B in a (C3-C3)-free plane graph G, we have 1. P v∈B |Rv| ≤ 3|V (B)| − 3 when B /∈ {B1, B2} or B ∼= B3 and v1v2v3 is not a face; 2. P v∈B |Rv| = 3|V (B)| when B ∼= B2 or B ∼= B1 and v1v3v4 is not a face or B ∼= B3 and v1v2v3 is a face; 3. P v∈B |Rv| = 3|V (B)| + 3 when B ∼= B1 and v1v3v4 is a face. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** for an example with n = 11 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The 3-face-blocks with P v∈B |Rv| > 3|V (B)|. Proof. The two properties are trivial, since the graph G is (C3-C4)-free. We give the proof of Lemma 4.3 by giving the partition of Rv for some vertex v ∈ V (B). Let (k1, k2, ..., kℓ) denote the partition of Rv. We use t to…

**Figure 5.** Figure 5: Relabel the vertices. Then we could assume that any two consecutive single 3-faces have no edge between them. If there exists a 3-face-block generated from the partition (1, 1, 1, 1, ...) which has the property that P u∈Rv |Ru| > 3|V (Rv)|, then we choose the graph tha…

**Figure 6.** Figure 6: Label the graph of B1. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: The extremal graph with n = 10. 5 Acknowledgments The authors would like to thank the anonymous referees for their valuable comments, which greatly improved the presentation of the results. The authors would also like to thank Luyi Li and Tong Li for helpful discussion…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 2 canonical work pages

[12]

P. Li. Planar Tur´ an number of the disjoint union of cycles.Discrete Applied Mathematics, 342:260–274, 2024

2024
[1]

Bondy and U

J. Bondy and U. Murty. Graph Theory. GTM No. 244, Springer, Berlin, 2008

2008
[2]

D. W. Cranston, B. Lidick´ y, X. Liu, and A. Shantanam. Planar Tur´ an numbers of cycles: a counterexample. Electronic Journal of Combinatorics, 29(3):Paper No. 3.31, 10, 08 2022

2022
[3]

C. Dowden. Extremal C4-free/C5-free planar graphs. Journal of Graph Theory, 83(3):213– 230, 2016

2016
[4]

Erd˝ os and A

P. Erd˝ os and A. H. Stone. On the structure of linear graphs. Bulletin of the American Mathematical Society, 52(12):1087–1091, 1946

1946
[5]

L. Fang, H. Lin, and Y. Shi. Extremal spectral results of planar graphs without vertex- disjoint cycles. Journal of Graph Theory , 106(3):496–524, 2024

2024
[6]

Ghosh, E

D. Ghosh, E. Gy˝ ori, R. R. Martin, A. Paulos, and C. Xiao. Planar Tur´ an number of the 6-cycle. SIAM Journal on Discrete Mathematics , 36(3):2028–2050, 2022. 17

work page 2022
[7]

Gy˝ ori, K

E. Gy˝ ori, K. Varga, and X. Zhu. A new construction for the planar Tur´ an number of cycles. Graphs and Combinatorics , 40(6):124, 2024

2024

Show all 15 references

[8]

Gy˝ ori, A

E. Gy˝ ori, A. Li, and R. Zhou. The planar Tur´ an number of the seven-cycle, arxiv: 2307.06909, 2023

2023 arXiv
[9]

Y. Lan, Y. Shi, and Z.-X. Song. Extremal H-free planar graphs. Electronic Journal of Combinatorics, 26(2):No. 2.11, 17, 2019

2019
[10]

Y. Lan, Y. Shi, and Z.-X. Song. Planar Tur´ an numbers of cubic graphs and disjoint union of cycles. Graphs and Combinatorics , 40(2):No. 28, 2024

2024
[11]

Lan and Z.-X

Y. Lan and Z.-X. Song. An improved lower bound for the planar Tur´ an number of cycles, arxiv: 2209.01312, 2022

2022 arXiv
[13]

R. Shi, Z. Walsh, and X. Yu. Dense circuit graphs and the planar Tur´ an number of a cycle. Journal of Graph Theory , 108(1):27–38, 2025

2025
[14]

R. Shi, Z. Walsh, and X. Yu. Planar Tur´ an number of the 7-cycle. European Journal of Combinatorics, 126:104134, 2025

2025
[15]

P. Tur´ an. On an extremal problem in graph theory. Matematikai ´ es Fizikai Lapok, 48:436–452, 1941. 18

1941

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

P. Li. Planar Tur´ an number of the disjoint union of cycles.Discrete Applied Mathematics, 342:260–274, 2024

2024

[2] [1]

Bondy and U

J. Bondy and U. Murty. Graph Theory. GTM No. 244, Springer, Berlin, 2008

2008

[3] [2]

D. W. Cranston, B. Lidick´ y, X. Liu, and A. Shantanam. Planar Tur´ an numbers of cycles: a counterexample. Electronic Journal of Combinatorics, 29(3):Paper No. 3.31, 10, 08 2022

2022

[4] [3]

C. Dowden. Extremal C4-free/C5-free planar graphs. Journal of Graph Theory, 83(3):213– 230, 2016

2016

[5] [4]

Erd˝ os and A

P. Erd˝ os and A. H. Stone. On the structure of linear graphs. Bulletin of the American Mathematical Society, 52(12):1087–1091, 1946

1946

[6] [5]

L. Fang, H. Lin, and Y. Shi. Extremal spectral results of planar graphs without vertex- disjoint cycles. Journal of Graph Theory , 106(3):496–524, 2024

2024

[7] [6]

Ghosh, E

D. Ghosh, E. Gy˝ ori, R. R. Martin, A. Paulos, and C. Xiao. Planar Tur´ an number of the 6-cycle. SIAM Journal on Discrete Mathematics , 36(3):2028–2050, 2022. 17

work page 2022

[8] [7]

Gy˝ ori, K

E. Gy˝ ori, K. Varga, and X. Zhu. A new construction for the planar Tur´ an number of cycles. Graphs and Combinatorics , 40(6):124, 2024

2024