Planar Tur\'{a}n numbers of three configurations

Ping Li; Xuqing Bai; Zhipeng Gao

REVIEW 5 major objections 6 minor 14 references

Planar Tur\'{a}n numbers of three configurations

T0 review · 5 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves that the planar Turán number of the disjoint union of a triangle and a Θ4 is floor(5n/2)−4 for all n≥174, and classifies every extremal graph; it also settles tight bounds for two related triangle–theta configurations.

desk verdict Completes the C3/Θ4 planar Turán family with plausible but not machine-checked finite classifications; Theorem 1.1's sharpness claim overreaches its construction. read the letter →

arxiv 2509.06131 v2 pith:I4FFLUOM submitted 2025-09-07 math.CO

Xuqing Bai , Zhipeng Gao , Ping Li This is my paper · ORCID

classification math.CO MSC 05C35

keywords planarTuránnumberextremalgraphsC3∪Θ4triangularblockstriangle-densitylinearforestthetaouterplanartriangle-free

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 establishes exact planar Turán numbers—the maximum number of edges in an n-vertex planar graph avoiding a fixed forbidden subgraph—for three configurations built by combining a triangle with a theta graph on four vertices, meaning two vertices joined by three internally disjoint paths. Its main result is that the disjoint union C3∪Θ4 attains floor(5n/2)−4 edges for every n≥174, with all extremal graphs explicitly described: essentially a K2 joined to a maximum matching, plus two variants for odd n and one cone over a triangle-free outerplanar graph. It also proves a tight upper bound of 13n/5−26/5 for K1+(P2∪P3), and a bound of floor(5n/2)−4 for the remaining configuration H5, with extremal characterizations. These results complete the planar Turán picture for all six ways to combine C3 and Θ4, a class that had only been partially resolved. Exact values and extremal characterizations are rare for small forbidden planar graphs, so this turns a numerical extremal question into a structural description.

What carries the argument

The paper's main structural tool is the triangular-block decomposition: partition all inner triangular faces into classes connected through shared edges, then assemble them into triangular components. A solid triangular block is one whose triangular holes have been filled, and the triangle-density ρ(B) counts true 3-faces per vertex. The first two theorems come from classifying all H4-free and H5-free solid triangular blocks—a finite list plus wheels and fans for H5—and proving a density bound ρ(D)≤(6|D|−12)/(5|D|) for H4-free components. For Theorem 1.3, the key object is EI(G), the set of edges lying on two triangular faces; each such edge generates a Θ4 subgraph, and the paper proves that

What would settle it

A concrete check: enumerate all H4-free solid triangular blocks up to order 9; if any is not isomorphic to a block in Figure 3, Lemma 3.2 fails and the 13n/5 upper bound collapses. For Theorem 1.3, a plane graph with n≥174 and more than floor(5n/2)−4 edges that is C3∪Θ4-free would falsify it directly.

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Extended reading notes

Core claim

The central claim is that the last unresolved configurations built from a triangle and a Θ4 have tight planar Turán numbers. For the disjoint union C3∪Θ4, the paper proves that for n≥174 the maximum is floor(5n/2)−4, and that every graph attaining it belongs to one of three explicit families: when n is even, a copy of K2 joined to a matching on the remaining n−2 vertices; when n is odd, either K2 joined to a near-perfect matching, a variant in which a new vertex is attached to endpoints of two matching edges, or a single universal vertex joined to a triangle-free outerplanar graph with the maximum possible number of edges. For K1+(P2∪P3), the paper proves the upper bound 13n/5−26/5 for n≥72

Load-bearing premise

The load-bearing premise is that the finite list of solid triangular building blocks in Figure 3 is complete for H4-free graphs; it is established by induction with case checks at small orders, so a missed block would break the 13n/5 bound.

Editorial extensions

If this is right

For n≥174, an n-vertex planar graph with no copy of C3∪Θ4 can have at most floor(5n/2)−4 edges, and this bound is attainable.
Every extremal C3∪Θ4-free graph for even n is exactly a copy of ((n−2)/2 K2)+K2, while odd n has three explicit extremal families.
The six C3-and-Θ4 combinations now all have determined or tightly bounded planar Turán numbers: three from earlier work and three from this paper.
For H5, extremal graphs are precisely those whose triangular components are copies of B5 or B′2, whose vertices are covered by those components, and whose faces are only 3-cycles or 4-cycles.
The bound for K1+(P2∪P3) is achieved by an explicit family with n=4k+2 vertices, so the upper bound is tight on that residue class.

Reading between the lines

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

Beyond the paper: the threshold n≥174 in Theorem 1.3 is likely not the true minimum; a finite exhaustive search for n<174 could lower it and reveal small exceptional extremal graphs.
Beyond the paper: the K2-joined-to-a-matching extremal shape suggests that similar unions of a triangle with a larger theta graph Θk will have planar Turán number floor(5n/2)−O(1) for large n, with analogous extremal families.
Beyond the paper: the triangular-block density argument should generalize to K1+(P2∪Pk) and other small linear forests, giving explicit linear upper bounds of the same type and identifying the residue classes where they are tight.
Beyond the paper: since 10x+6y is even and, for x≥2, represents every sufficiently large even integer, equality in Theorem 1.2 actually holds for all large even n; the genuinely open cases are odd n and a finite set of small even n.

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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

Quick take: this is a solid, workmanlike paper that closes out a natural six-case family. It resolves the three remaining C3/Θ4 configurations, gives exact bounds in all three, and in two cases (H5, H6) characterizes all extremal graphs. The triangular-block machinery is extended carefully, and the case analyses are detailed enough that a patient referee can check them. The H6 proof adapts Li's EI(G) technique cleanly; the self-citation there is for a borrowed tool, not a circular crutch.

The soft spots are localized but real. Theorem 1.1 states equality for all n ≡ 2 mod 5, but the only construction shown achieves n ≡ 2 mod 20. That is a genuine gap in the lower bound; either the statement needs to be narrowed or the authors need to supply constructions for the other residue classes. Section 3 also mislabels H4 as K1 + (P2 ∪ P4); it should be P2 ∪ P3. Both are fixable.

The bigger worry is one the reader should carry into refereeing: the finite classifications—Lemma 3.2 for H4-free triangular blocks, Observation 5.2 for overlapping Θ4s, and the ensuing claims in Lemma 5.3—are the load-bearing parts, and they are not machine-checked. Observation 5.2 is asserted with a figure and no proof. If it misses a mixed-overlap configuration, the bound |EI(G)| ≤ n/2 + 4 and the extremal family in Theorem 1.3 could change. This is not a manufactured flaw; it is the nature of this kind of proof. The cases are small enough that a referee can verify them by hand, but it will take real work.

Bottom line: the paper deserves a serious referee. The core program is coherent, the results are new, and the local issues are correctable. I would not desk-reject. If I were editor, I'd ask for a revision that fixes the Theorem 1.1 sharpness statement, corrects the H4 typo, and preferably expands the proof of Observation 5.2 or at least flags it as a claim needing verification.

Referee Report

5 major / 6 minor

Summary. The paper studies planar Turán numbers for three small graphs obtained by combining C3 and Θ4. Theorem 1.1 gives the upper bound ex_P(n,K1+(P2∪P3)) ≤ 13n/5 − 26/5 for n ≥ 72, with equality claimed for all n ≡ 2 (mod 5). Theorem 1.2 gives the upper bound ex_P(n,H5) ≤ floor(5n/2) − 4, exact for n representable as 10x+6y with x≥2, y≥0, together with a characterization of equality via Remark 1. Theorem 1.3 claims ex_P(n,C3∪Θ4) = floor(5n/2) − 4 for n ≥ 174 and fully characterizes the extremal graphs: for even n, a join of K2 with a matching; for odd n, either K2+M_{n−2}, K2∨M_{n−2}, or u+O with O a C3-free outerplanar graph of even order. The proofs use a decomposition into triangular blocks and triangular components, density inequalities, and extensive local case analyses.

Significance. If the results are correct, the paper makes a strong contribution to the planar Turán problem for small unions of triangles and Theta graphs: it provides exact bounds for three configurations and, in Theorem 1.3, a full extremal-graph characterization for n ≥ 174. The high-level method—decomposing the graph into triangular blocks, bounding triangle densities, then deriving an edge bound—is appropriate and does not involve fitted parameters or circular reasoning. The main value depends, however, on several finite local classifications that are currently asserted rather than fully demonstrated, and on a lower-bound construction that does not meet the stated residue condition for Theorem 1.1.

major comments (5)

[§3, tightness paragraph after proof of Theorem 1.1] The lower-bound construction G_k is said to have |G_k|=4k+2 and e(G_k)=13n/5−26/5. For this number to be an integer (and hence to equal the stated upper bound), one needs 5 | 4k, i.e. n ≡ 2 (mod 20). Theorem 1.1 asserts equality for every n ≡ 2 (mod 5), including n=72, which is not of the form 4k+2 with k≥14. No other extremal family is supplied. Thus the equality statement is not established as written. Either add constructions covering all n ≡ 2 (mod 5) or restrict the theorem.
[§3, Lemma 3.2 and proof of Lemma 3.3] The classification of H4-free solid triangular blocks is load-bearing for Table 1 and Lemma 3.3. The proof handles |B|=6,7,8 by phrases such as 'then B is isomorphic to...' without enumerating the possible ways to add a vertex to each predecessor, and Lemma 3.3 uses 'one can verify' for several small configurations in Figure 4. Because Proposition 3.1 only guarantees one removable boundary vertex, a missed extension at order 8 would propagate to all larger orders. Please provide the full finite case check, or a machine-checkable enumeration.
[§5, Observation 5.2 and Lemma 5.3] The proof of Theorem 1.3 rests on the local classification of Θe∪Θf as D1,D2,D3 and on the subsequent claims that every g∈Be forms D1 with e, that Be is a matching, that |B*_e|≤5, and that Θf∪Θg is D1 in Case 2.2. These are asserted with 'otherwise ... a contradiction' but the local case analysis is not shown. If a mixed-overlap configuration exists (e.g. Θg sharing {x,a} with Θe) or |B*_e|=6 is possible, the sharp bound |EI(G)|≤⌊n/2⌋+4 and the extremal characterization in Lemma 5.3 fail. This is the core gap; the authors should supply a complete enumeration or a formal proof.
[§5, Lemma 5.3, first paragraph] The assertion 'Ae must contain at least 4 independent edges including f' for each e and f∈Ae is not justified. A graph of maximum degree at most 9 with many edges need not contain a matching of size 4 containing a prescribed edge. The later arguments choose an edge h disjoint from a specified 2- or 3-set based on this claim. Please prove it or replace the counting.
[§5, Claim 5.5] In the proof that Be is a matching, the displayed contradiction 'Θxb ∪ ypqy is a copy of C3∪Θ4' is not verifiable as written: in the D1 configuration shown in Figure 9(a), the edge xb is not known to exist. If this is a typo for another Θ-graph, it needs correction; otherwise the matching property of Be is unsupported. The bound |Be|≤(n−2)/2 depends on it.

minor comments (6)

[§3, first paragraph] H4 is defined as K1+(P2∪P3), but the section opens with 'H4 = K1+(P2∪P4)'. Please correct this mislabel.
[§1 and abstract] The abstract says previous work solved cases when L is a path or a matching, or satisfies |L|≥7; the introduction does not state the |L|≥7 result or give a reference. Also, 'for 3 ≤ k ≤ 6' should be 'for 3 ≤ t ≤ 6' or similar.
[§1 and §5] The symbol '+' is used both for disjoint union in the abstract (P2∪P3) and for join in Theorem 1.3 (M_{n-2}+K2). Define '+' explicitly in the notation section to avoid ambiguity with the disjoint-union symbol ∪.
[§5, Claim 5.7] The inequality f3(G) ≤ n−1 after showing every 3-face contains u is asserted without the incidence argument; the short proof (each non-u vertex can be in at most two such 3-faces) should be stated.
[§4 and acknowledgments] There are typos: 'Through inductively construction' should be 'Through induction', and 'Ping Li is suppose by' should be 'supported by'.
[§3, Figure 5 caption] The caption says 'Theorem 1' but should refer to 'Theorem 1.1'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the planar Turán bounds are derived from structural classifications, with no fitted parameters or predictions defined in terms of their targets.

full rationale

The paper's chain is self-contained. Theorems 1.1–1.3 are obtained from explicit structural lemmas (Propositions 3.1 and 4.1, Lemmas 3.2, 3.3, 4.3, 5.3) that classify H-free triangular blocks and bound triangle densities; these are finite enumerations and inductive arguments, not fits. The equality cases are characterized from the same structural conditions (e.g., Remark 1 and the equality analysis in the proof of Theorem 1.3), so no 'prediction' reduces to an input by construction. The only self-citation, [12], is used for the idea of analyzing |EI(G)|; the actual lemmas in Section 5 are proved in the paper, so the citation is not load-bearing. The compressed local case checks (e.g., Observation 5.2 and the 'otherwise ... contradiction' steps) are potential correctness risks, but a finite case enumeration that might be incomplete is not circularity under the given rubric.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard graph theory, on prior external Turán results, and on the paper's own structural decomposition into triangular blocks and components. No fitted constants or new postulated objects are introduced. The main burden is the correctness of the finite classification lemmas, which is an internal proof obligation rather than an external axiom.

assumptions (4)

standard math Euler's formula and the face counting identity 2e = sum i f_i
Used throughout Sections 3 to 5 to convert triangle-count bounds into edge bounds, e.g., equations (2)-(4) in the proof of Theorem 1.1.
domain assumption Prior result ex_OP(m,C3) = (3m-4)/2 for even m (Fang and Zhai [4])
Used in Theorem 1.3 to identify C3-free outerplanar graphs of even order and to measure the extremal graph {u}+O.
domain assumption Prior planar Turán values for C4, C5, K1+2P2, K1+3P2, and K1+P_t (Dowden [1], Fang-Wang-Zhai [3], Lan-Shi-Song [9])
Used to state known values for H1 and H2 and as external context. These results are not used to derive the new bounds.
domain assumption The EI(G) and pseudo-face technique from Li [12] is applicable to C3∪Θ4-free planar graphs
Section 5 says the proof follows the idea proposed in [12]. This is a borrowed method, not a circular input, but the new proof depends on its correctness in this setting.

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Pith. "Pith review of Planar Tur\'{a}n numbers of three configurations." pith.science (2026). https://pith.science/paper/I4FFLUOM

@misc{pith2026250906131,
  author       = {Pith},
  title        = {Pith review of: Planar Tur\'an numbers of three configurations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I4FFLUOM}},
  note         = {Machine review of arXiv:2509.06131}
}

abstract

The planar Tu\'{a}n number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for example, $H$ is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, $\Theta$-graphs and other small graphs formed by a union of them. One representative graph among such union graphs is $K_1+L$ where $L$ is a linear forest without isolated vertices. Previous works solved the cases when $L$ is a path or a matching. In this work, we first investigate the planar Tur\'{a}n number of the graph $K_1+L$ when $L$ is the disjoint union of a $P_2$ and $P_3$. Equivalently, $K_1+L$ represents a specific configuration formed by combining a $C_3$ and a $\Theta_4$. We further consider the planar Tur\'{a}n numbers of the all graphs obtained by combining $C_3$ and $\Theta_4$. Among the six possible such configurations, three have been resolved in earlier works. For the remaining three configurations (including $K_1+(P_2\dot{\cup}P_3)$), we derive tight bounds. Furthermore, we completely characterize all extremal graphs for the remaining two of these three cases.

Figures

Figures reproduced from arXiv: 2509.06131 by the authors.

**Figure 1.** Six types of combinations of C3 and Θ4. exP (n, H1) = 3n − 6 when n ≥ 6. Fang, Wang, and Zhai [3] showed that exP (n, H2) ≤ 8 3 (n − 2) with a sharp upper bound. Because H1 is a subgraph of H3, exP (n, H3) = 3n − 6 follows directly. In addition to H4, we also focus on determining the planar Tur´an numbers of the remaining two graphs H5 and H6. For H5, we determine its planar Tur´an number exactly when n = 10x + 6y h… view at source ↗

**Figure 2.** An example on TBs and solid TBs. 3 Proof of Theorem 1.1 This section focuses on the planar Tur´an number of H4 = K1 + (P2∪˙ P4). We begin with a proposition on H4-free solid TBs, followed by a characterization of all such solid TBs. Proposition 3.1. Let B be an H4-free solid TB with |B| ≥ 4 and outer boundary C. Then there exists a vertex v ∈ V (C) such that B − v is a solid TB of order |B| − 1, unless B ∼= B (n) 15… view at source ↗

**Figure 3.** H4-free triangular blocks, with dashed circles indicating potential junction vertices. |B (n) i | = n for i ∈ {11, 12, 13, 14, 15}. Lemma 3.2. Every H4-free solid TB is isomorphic to a configuration in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: all possible configurations of D′ when |D′ | ≤ 6, where the gray area denotes the hole. If H ≇ B10, then ∆H ≤ |H| − 2. Thus, ρ(D) = ∆D′ + ∆H |D| ≤ 6|D′ |−12 5 + |H| − 2 |D| = 6|D| − 12 − (|H| − 2) 5|D| < 6|D| − 12 5|D| . If H ∼= B10, then ρ(D) = ∆D′ + 6 |D′ | + 5 ≤ 6|D…

**Figure 5.** Figure 5: The extremal graph Gk. 4 Proof of Theorem 1.2 This section is dedicated to studying the planar Tur´an number of H5. We begin by introducing a structural property of H5-free solid TBs, then fully characterize these solid TBs, and finally determine the triangle-densities…

**Figure 6.** Figure 6: A part of H5-free triangle-blocks. For |B| = 7, B − v is a solid TB belonging to {B6, B′ 1 , B′ 2 , W5, F5}. If B − v is a copy in {B6, B′ 2 , W5}, then B contains H5 as a subgraph, a contradiction; If B − v is a copy of B′ 1 , then B ∼= B′ 3 ; If B − v is a copy of F5…

**Figure 7.** Figure 7: All possible TBs of B′ . Cases B′ 1 B′ 2 B′ 3 Wk Fk+1 ∆B(B ∈ {B′ i (i ∈ [3]), Wk, Fk+1) ≤ 5 6 6 k k Triangle density ≤ 5 6 1 6 7 k k+1 k k+2 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: The graph R′ k 5 Proof of Theorem 1.3 In this section, we study the planar Tur´an number of C3∪˙ Θ4. Let G be a C3∪˙ Θ4-free plane graph with |G| ≥ 174. A set of edges is called independent edges if they form a matching. The proof of Theorem 1.3 could be proceeded usin…

**Figure 9.** Figure 9: The planar structures constituted by Θe ∪ Θf are D1, D2, and D3. Specifically, D1 contains two holes, namely auyva and xubvx. D2 contains two holes, which are xvyax and xuybx. And D3 contains two holes, axua and buyvxb. Let {e1, · · · , et} ⊆ EI (G). For any inner face…

**Figure 10.** Figure 10: The plane graphs D1,1 and D1,2. Case 2.2. For every edge e in EI (G) and f ∈ Ae, Θe ∪ Θf is a copy of D3. Since Ae contains at least 4 independent edges, we choose two of them, say f, g. It is clear that Θe ∪ Θf and Θe ∪ Θg are copies of D3. Hence, Θf ∪ Θg is a copy o…

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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work page 2022

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work page 2023

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Planar Tur´ an number of the 6-cycle

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Yongxin Lan and Yongtang Shi. Planar Tur´ an numbers of short p aths. Graphs and Combina- torics, 35(5):1035–1049, 2019

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