{"id":"28cb4ea0-6970-4e77-a6ac-393ca07d614e","arxiv_id":"2509.06131","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact planar Turán numbers are established for K1+(P2∪P3), a combined C3/Θ4 configuration, and the disjoint union of C3 and Θ4, with extremal graph characterizations.","lead":"This paper determines the maximum number of edges a planar graph can have while avoiding three specific small forbidden subgraphs, and describes the extremal graphs in two cases. It completes the study of all six graphs obtained by combining a triangle and a theta graph.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 rests on Lemma 5.3's unverified finite classification of overlapping Θ4s; an error there would change the extremal family.","rationale":"I read Theorem 1.3 as the paper's central claim. Its upper bound is a weighted count driven by |EI(G)|, and every non-routine step is Lemma 5.3. The proof of Lemma 5.3 is a finite but compressed enumeration of how two Θ4s can overlap in a planar C3∪Θ4-free graph. The paper marks several points as 'it is easy to verify' or 'a contradiction,' but these are exactly the load-bearing checks on which the sharp 1-slack bound and the extremal classification depend. The reader's weakest_assumption selected Lemma 3.2, a similar enumeration for the H4 theorem; that is a reasonable concern for Theorem 1.1, but it is not the principal support for the headline result. My concern is not about consistency with prior consensus; it is internal completeness: the locality of Lemma 5.3 is finite and checkable, and the proof's abbreviated 'otherwise' steps do not, by themselves, certify it. If the enumeration is correct, Theorem 1.3 stands; if it is not, the extremal family could change. I therefore maintain the CONDITIONAL verdict, agreeing with the reader's overall posture but shifting the focus to Section 5.","tokens_in":21823,"tokens_out":34966,"duration_ms":350575,"concrete_test":"Run an exhaustive computational check of the local configurations used in Lemma 5.3. Generate all planar graphs on 6 vertices containing two diamonds Θe,Θf with e and f independent and both edges in EI (up to isomorphism, via nauty/geng plus a planarity test), and verify Observation 5.2. Then, for every Case-1 configuration, enumerate all possible additional EI edges g on the same vertex set and check Claims 5.4–5.6: (i) every g∈Be with Θe∪Θg not isomorphic to D1 creates C3∪Θ4; (ii) xy∉E(G); (iii) |B*_e|≤5. Repeat with 7- and 8-vertex local augmentations, allowing ΔI≤9. If all checks pass, Lemma 5.3's enumeration is certified; if any configuration violates them, the extremal characterization in Theorem 1.3 needs revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is Theorem 1.3, and its proof is carried by Lemma 5.3. The entire argument depends on Observation 5.2: that for independent EI-edges e,f, the plane subgraph Θe∪Θf is exactly one of D1, D2, D3. The proof then asserts several follow-up facts without displaying the underlying case analysis: in Case 1, that every g∈Be also forms D1 with e (Claims 5.4–5.5), that xy∉E(G), and that |B*_e|≤5 (Claim 5.6); in Case 2.2, that Θf∪Θg is a copy of D1. These are finite, local statements, but the text compresses them into 'otherwise ... a contradiction' lines. If the true local classification permits a mixed-overlap configuration (for example, Θg sharing {x,a} with Θe) or permits |B*_e|=6 via a K4 on V(Θe), then the sharp bound |EI(G)|≤⌊n/2⌋+4 and the equality-case characterization in Lemma 5.3 could fail. Since Theorem 1.3 builds its extremal classification precisely on that equality case, an error here would not just weaken a numerical bound; it would change the list of extremal graphs. The reader's weakest_assumption focused on Lemma 3.2, which supports Theorem 1.1; for the paper's headline result, the analogous unverified enumeration is Lemma 5.3, and it is not independently checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22192,"tokens_out":15236,"duration_ms":165836,"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":[{"comment":"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.","section":"§3, tightness paragraph after proof of Theorem 1.1"},{"comment":"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.","section":"§3, Lemma 3.2 and proof of Lemma 3.3"},{"comment":"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.","section":"§5, Observation 5.2 and Lemma 5.3"},{"comment":"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.","section":"§5, Lemma 5.3, first paragraph"},{"comment":"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.","section":"§5, Claim 5.5"}],"minor_comments":[{"comment":"H4 is defined as K1+(P2∪P3), but the section opens with 'H4 = K1+(P2∪P4)'. Please correct this mislabel.","section":"§3, first paragraph"},{"comment":"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.","section":"§1 and abstract"},{"comment":"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 ∪.","section":"§1 and §5"},{"comment":"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.","section":"§5, Claim 5.7"},{"comment":"There are typos: 'Through inductively construction' should be 'Through induction', and 'Ping Li is suppose by' should be 'supported by'.","section":"§4 and acknowledgments"},{"comment":"The caption says 'Theorem 1' but should refer to 'Theorem 1.1'.","section":"§3, Figure 5 caption"}],"recommendation":"major_revision","confidential_remarks":"I see no circularity or parameter fitting; the concerns are about unverified finite enumerations and one mismatch between a construction and a stated residue class. These are fixable in principle but the paper cannot be accepted until Lemma 3.2 and Lemma 5.3 are backed by complete case analyses or machine-checked code, and the lower-bound issue in Theorem 1.1 is resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nBottom 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.","headline":"Completes the C3/Θ4 planar Turán family with plausible but not machine-checked finite classifications; Theorem 1.1's sharpness claim overreaches its construction.","tokens_in":22634,"tokens_out":4264,"would_cite":true,"duration_ms":44167,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["planar Turán number","extremal planar graphs","C3 ∪ Θ4","triangular blocks","triangle-density","linear forest","theta graphs","outerplanar triangle-free graphs"],"falsifier":"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.","tokens_in":21741,"feed_emoji":"📐","tokens_out":10388,"duration_ms":106840,"temperature":0.7,"pith_summary":"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.","feed_headline":"Disjoint triangle and theta graph maxes at 5n/2−4 edges","feed_subtitle":"Large-n extremal graphs are explicitly classified; two sibling configurations are also settled.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Launched the planar Turán problem and supplies the exact bound for the triangle-only configuration H1.","marker":"[1]"},{"why":"Determined preceding tight bounds for intersecting triangles, including H2, and established K1+linear-forest results that this paper extends.","marker":"[3]"},{"why":"Gives the outerplanar triangle-free edge count ex_OP(n,C3) used in the odd-n extremal description for Theorem 1.3.","marker":"[4]"},{"why":"Supplies the general upper-bound framework for K1+H when H is a linear forest, which Theorem 1.1 sharpens for H=P2∪P3.","marker":"[9]"},{"why":"Provides the extremal theory of theta-free planar graphs needed for the Θ4 side of the configurations.","marker":"[10]"},{"why":"Introduces the EI(G) edge-counting method for disjoint cycles that Theorem 1.3 adapts to handle C3∪Θ4.","marker":"[12]"}],"fun_headline_variants":["Triangle-theta planar Turán bound: floor(5n/2)-4 for n≥174","C3∪Θ4 max edges in planar graphs: floor(5n/2)-4","Extremal graphs classified for disjoint triangle-theta configuration","Tight planar Turán bounds for K1+(P2∪P3) and triangle-theta unions","Solving the last triangle-theta planar Turán cases exactly"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Triangle-theta planar Turán bound: floor(5n/2)-4 for n≥174","C3∪Θ4 max edges in planar graphs: floor(5n/2)-4","Extremal graphs classified for disjoint triangle-theta configuration","Tight planar Turán bounds for K1+(P2∪P3) and triangle-theta unions","Solving the last triangle-theta planar Turán cases exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001019,"raw_usage":{"total_tokens":4195,"prompt_tokens":857,"completion_tokens":3338,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":601,"completion_tokens_details":{"reasoning_tokens":3230}},"tokens_in":601,"tokens_out":3338,"duration_ms":25935,"temperature":1.0,"reasoning_tokens":3230,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:21:28.055717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Extremal C4-free/C5-free planar graphs","cited_arxiv_id":null,"evidence_quote":"Launched the planar Turán problem and supplies the exact bound for the triangle-only configuration H1."},{"cited_title":"Planar Tur´ an numbe r of intersecting triangles","cited_arxiv_id":null,"evidence_quote":"Determined preceding tight bounds for intersecting triangles, including H2, and established K1+linear-forest results that this paper extends."},{"cited_title":"Outerplanar tur´ an numbers of cycles and paths","cited_arxiv_id":null,"evidence_quote":"Gives the outerplanar triangle-free edge count ex_OP(n,C3) used in the odd-n extremal description for Theorem 1.3."},{"cited_title":"Extremal H-free planar graphs","cited_arxiv_id":null,"evidence_quote":"Supplies the general upper-bound framework for K1+H when H is a linear forest, which Theorem 1.1 sharpens for H=P2∪P3."},{"cited_title":"Extremal Theta-f ree planar graphs","cited_arxiv_id":null,"evidence_quote":"Provides the extremal theory of theta-free planar graphs needed for the Θ4 side of the configurations."},{"cited_title":"Planar Tur´ an number of the disjoint union of cycles","cited_arxiv_id":null,"evidence_quote":"Introduces the EI(G) edge-counting method for disjoint cycles that Theorem 1.3 adapts to handle C3∪Θ4."}],"review_version":1}