{"id":"3b92042b-3a32-42ad-8518-768b08c249c7","arxiv_id":"2411.18487","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The exact planar Turán numbers are determined for the graphs C3-C3 and C3-C4 (two disjoint cycles joined by an edge).","lead":"This paper determines the largest possible number of edges in a planar graph that avoids two small forbidden patterns: two triangles joined by an edge (C3-C3), and a triangle joined to a 4-cycle (C3-C4). A generalist might care because it resolves two open extremal graph questions and corrects a claimed value for two disjoint triangles in a prior paper.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.3's t=0 case (Case 4.5) does not establish the block-surplus bound it claims, and the C3-C4 upper bound depends on that classification.","rationale":"The reader's weakest-assumption analysis identified the exceptional-block classifications in Lemma 3.3 and Lemma 4.3 as the discharging premise on which the upper bounds rest. My stress-test confirms this and sharpens it: Lemma 4.3's Case 4.5 does not actually prove the required inequality. The text switches from a local block analysis to a global statement '3f_3(G) = e_3 <= e' and then asserts the block bound without a connecting argument. This is not merely a skipped 'similar argument'; it is an apparent logical gap in a central lemma. Because the proof of Theorem 1.2 uses Lemma 4.3 to limit the number and type of bad blocks, the final counting step depends on this classification. I am not claiming the theorem is false: the extremal constructions and the overall strategy are plausible, and the reader's conditional verdict is appropriate. However, the paper should not be accepted until Case 4.5 is repaired or independently verified. I also note the Corollary 1.2 inconsistency flagged by the reader, but I treat it as a separate correctness issue: it does not bear on the upper-bound proof of Theorem 1.2 as directly as the missing Case 4.5 argument does.","tokens_in":17412,"tokens_out":18302,"duration_ms":170880,"concrete_test":"Run an exhaustive enumeration of all plane graphs on up to 12 vertices, filter for C3-C4-free graphs, decompose them into 3-face-blocks, and check whether any block with a partition (1,1,...,1) of some R_v satisfies P |R_u| > 3|V(B)|. If such a block exists, Lemma 4.3 is false; if none exists, the conclusion of Case 4.5 is true but still needs a proof that does not use the global e_3 <= e identity. A supplementary check is to re-derive Case 4.5 from local counts, replacing the final 'e_3 <= e' step with an explicit bound on the number of 3-faces incident with the block.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.2 is proved by reducing the upper bound to Lemma 4.3, which asserts that every 3-face-block B of a (C3-C4)-free plane graph satisfies P_{v in B} |R_v| <= 3|V(B)| except for four listed blocks. The proof of Lemma 4.3 is therefore load-bearing: if any unlisted block has surplus, the later counting argument and the final bound e(G) <= floor(5n/2)-4 do not follow. The weakest point in that proof is Case 4.5, which treats blocks whose chosen R_v has partition (1,1,...,1) with |R_v| >= 4. After a minimal-counterexample argument, the text states: 'Hence we have e_{3,3} = 0 and then 3f_3(G) = e_3. According to Property 2.1 we have 3f_3(G) = e_3 <= e. Then for t = 0, we have P_{u in R_v} |R_u| <= 3|V(R_v)|.' This is a non sequitur: the inequality e_3 <= e is a global identity about facial edge counts and does not bound the surplus of a single block. It gives at most f_3(G) <= e/3 for the whole graph, which is compatible with a block having P |R_u| > 3|V(B)| when other blocks have compensating slack. The conclusion that the t=0 case satisfies the block inequality is therefore not proved by the text. Since Theorem 1.2 explicitly relies on Lemma 4.3's classification, this gap is a genuine load-bearing concern: a missing exceptional block with surplus would break the case split m >= k1+k2 and the subsequent Euler-formula bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17686,"tokens_out":14326,"duration_ms":120584,"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":[{"comment":"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.","section":"Section 4, Lemma 4.3, Case 4.5"},{"comment":"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.","section":"Introduction and Corollary 1.2"},{"comment":"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.","section":"Lemmas 3.3 and 4.3"}],"minor_comments":[{"comment":"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":"Section 4, Lemma 4.3, Figure 4"},{"comment":"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":"Section 4, proof of Theorem 1.2"},{"comment":"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.","section":"Section 4, Lemma 4.3"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural problem and the overall approach is plausible, but I would not accept it in its current form. The Case 4.5 gap in Lemma 4.3 is a genuine proof gap in the upper bound for Theorem 1.2, and the Corollary 1.2 inconsistency must be fixed; the latter may be a simple typo, but the former requires real additional argument. I recommend major revision rather than rejection because the central claims may survive once the case analysis is completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nSecond, 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.\n\nThird, 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.\n\nBottom 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.","headline":"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.","tokens_in":18248,"tokens_out":4569,"would_cite":false,"duration_ms":37006,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $n=7$…","keywords":["planar Turán number","adjacent cycles","C3-C3","C3-C4","3-face-block","discharging method","extremal graphs","disjoint cycles"],"falsifier":"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.","tokens_in":17160,"feed_emoji":"🔺","tokens_out":7546,"duration_ms":55911,"temperature":0.7,"pith_summary":"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 $n=6$.","feed_headline":"Exact planar Turán numbers for two linked cycles","feed_subtitle":"C3-C3-free graphs cap at ⌈5n/2⌉-5 edges for n≥7; C3-C4-free at ⌊5n/2⌋-4.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Gives the planar Turán number of 2C3 with a small-n error that this paper corrects; the C3-C3 result extends and corrects it.","marker":"[10]"},{"why":"Gives ex_P(n, C3∪C4) for n≥20; this paper extends it to all n and uses the same C3-C4-free setting.","marker":"[12]"},{"why":"Establishes ex_P(n,tC)=3n-6 for t≥3, the maximal-planar baseline that the new formulas deviate from.","marker":"[9]"},{"why":"Initiated the planar Turán problem and provided the framework of counting faces that the proofs use.","marker":"[3]"}],"fun_headline_variants":["Linked cycles: planar Turan numbers now exact","C3-C3 and C3-C4: planar Turan numbers settled","Exact planar Turan numbers for two adjacent cycles","Planar Turan numbers for linked cycles: exact values","Two linked cycles: max planar edges determined"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Linked cycles: planar Turan numbers now exact","C3-C3 and C3-C4: planar Turan numbers settled","Exact planar Turan numbers for two adjacent cycles","Planar Turan numbers for linked cycles: exact values","Two linked cycles: max planar edges determined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000943,"raw_usage":{"total_tokens":4045,"prompt_tokens":981,"completion_tokens":3064,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":2985}},"tokens_in":597,"tokens_out":3064,"duration_ms":21715,"temperature":1.0,"reasoning_tokens":2985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:09:40.840994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}