{"id":"0cefb137-d482-4f6a-949e-46f600d2f453","arxiv_id":"2507.16351","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For n ≥ 295660, the planar Turán number of C3∪C5 is floor((8n-13)/3), and the unique extremal planar graph is described.","lead":"This paper determines the exact maximum number of edges in a large planar graph that contains no vertex-disjoint triangle and pentagon. The answer is a closed formula, and the unique graph that reaches the bound is identified.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The high-degree case of Lemma 3.4 is the pivotal unverified step: the fan-intersection and 'new vertex' arguments that force a 5-cycle avoiding v are not proved, and Theorem 1.1's upper bound depends entirely on this lemma.","rationale":"The reader's weakest-assumption identification of Lemma 3.4 is correct, and I agree with a CONDITIONAL verdict. The arithmetic in Lemma 3.1 and Claim 3.2 checks out, and the extremal construction is indeed C3∪C5-free because every cycle in K2 joined to copies of P3 contains one of the two K2 vertices. The bounded-degree case of Lemma 3.4 is not the main problem: if Δ≤105, a fixed 5-cycle C has at most 5+5·103=520 vertices in its radius-1 neighbourhood, so |B|≥521 yields a vertex u at distance 2, and a 3-face containing u is disjoint from C. The real load-bearing concern is the high-degree fan argument: it relies on nontrivial planarity assertions and on a 'new vertex y' step that is not proved. Since Lemma 3.5 and the final contradiction in Section 4 depend on Lemma 3.4, the proof is incomplete as written. I do not see circular reasoning or parameter fitting, and the result may well be true; hence CONDITIONAL is the appropriate verdict.","tokens_in":7078,"tokens_out":36214,"duration_ms":400046,"concrete_test":"Independently re-derive the high-degree case of Lemma 3.4 with explicit fan notation. Prove specifically: (i) a C5 avoiding v must meet each fan outside v; (ii) at most three fans can have ≥2 neighbours in C\\v; and (iii) when a second 3-face is added to one of the edges {u1x, wx, u2w}, its third vertex is outside {u1,u2,w,x} or, if it is not, a 5-cycle avoiding v still exists. A minimal computational check on all triangular blocks with ≤12 vertices and Δ≥6 would show whether the analogous 'new vertex' closure can fail in small cases; if it fails, the proof of Lemma 3.4 is invalid.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3.4 is the hinge of Theorem 1.1: Lemma 3.5 uses it to prove that in Cases (2) and (3) all 3-cycles pass through one vertex, and Section 4 then derives e ≤ 5n−9/2 < 8n−13/3. The bounded-degree half (Δ≤105) can likely be repaired by counting the radius-1 neighborhood of a 5-cycle, so the real gap is the high-degree half (Δ≥106). After partitioning the 3-faces incident to v into t≥8 fans, the proof asserts without derivation that any 5-cycle avoiding v must meet every fan, that at most three fans avoiding C\\v can have ≥2 neighbours in C\\v, and then claims that the second 3-face on one of {u1x, wx, u2w} must introduce a new vertex y, giving a 5-cycle u1u2wxy. None of these is justified from the definition of triangular block: the second 3-face could close on an existing vertex, and the illustrations (Figures 7–8) are not a proof. If Lemma 3.4 fails, Lemma 3.5 and the upper bound collapse; no alternative argument is given.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the planar Turán number of the disjoint union of a triangle and a 5-cycle. The main result (Theorem 1.1) states that for n ≥ 295660, ex_P(n, C3 ∪ C5) = floor((8n − 13)/3), with the unique extremal graph being K2 joined to floor((n − 2)/3) copies of P3 plus one additional short path depending on n mod 3. The proof proceeds by decomposing the graph into triangular blocks, using the Dowden bound for C5-free planar graphs to force many C5-containing blocks (Lemma 3.1), then showing structural restrictions on the blocks (Lemmas 3.2–3.5), and finally applying Euler's formula to bound the edge count.","tokens_in":7278,"tokens_out":25252,"duration_ms":260228,"significance":"If correct, the result would be a valuable addition to the emerging theory of planar Turán numbers for disjoint cycles, complementing the C3 ∪ C4 and 2C4 results. The extremal construction is clean and the lower bound is straightforward, and the paper gives an extremal characterization, which is stronger than most results in this area. The overall strategy—using Dowden's C5 benchmark and the triangular-block decomposition—is coherent and appropriate. However, the proof as written contains several unproved structural assertions at load-bearing points: the high-degree case of Lemma 3.4 and the block-classification claims in Lemma 3.1 need substantial repair before the upper bound is established. The manuscript does not provide machine-checked proofs or code, so the verification burden rests entirely on the written arguments.","major_comments":[{"comment":"The proof of the high-degree case is a chain of assertions with no justification. First, the claim that every 5-cycle must contain v, because otherwise a 5-cycle C* avoiding v would intersect at least one vertex of each fan F^i, is asserted but not proved; it is not evident why a 5-cycle avoiding v must meet every fan of 3-faces around v. Second, the statements that the extended 3-faces 'must intersect with C' and that planarity implies at most three fans have at least two neighbours in C \\ {v} are given without derivation. Most seriously, after finding the 3-face u1wx with x not in C, the proof says one of {u1x, wx, u2w} must belong to two 3-faces and 'let y be the new vertex'; the second 3-face could close on an existing vertex (u1, u2, w, or x), and no argument is provided that a new vertex is forced. Consequently, the asserted 5-cycle u1u2wxy need not have five distinct vertices. Since Lemma 3.4 underpins Lemma 3.5 and the entire upper-bound proof in Section 4, this gap is load-bearing.","section":"§3, Lemma 3.4"},{"comment":"The proof asserts that every triangular block that does not contain C5 and is not isomorphic to B2_4 must be one of B1_2, B1_3, B1_4. This is equivalent to asserting that every triangular block with at least five vertices contains a 5-cycle, but no proof is given, and the claim is not immediate from the definition of a triangular block. The inequality f3(G) ≤ (2/5)e(G) + 1037α + 30, which is essential for the contradiction in Claim 3.2, relies entirely on this classification. If a C5-free triangular block with five or more vertices exists, the ratio f3/e for such a block can be much larger than 2/5, and the resulting Euler-based contradiction would not follow. This missing classification is a second load-bearing gap in the proof of Lemma 3.1.","section":"§3, Claim 3.2 in Lemma 3.1"},{"comment":"The proof of Lemma 3.3 is only a few lines. It asserts that if B1 is not one of the listed blocks, then 'we could find a C5 in B1 that only intersects at most one vertex of {u,v}', and then 'we could find C3 ∪ C5 in B1 and B2'. The first assertion requires a structural result about triangular blocks that is not proved; Lemma 3.2 only applies to nonadjacent vertices lying in a hole of a block, and the manuscript does not verify that u and v (after deleting euv, if it exists) are nonadjacent vertices in a hole of B1. The second assertion also needs the existence of a triangle in B2 that is vertex-disjoint from the chosen C5, which is not automatic. Since Lemma 3.3 is used in Section 4 to derive the bound e(G') ≤ (8n − 16)/3 in Case (1) of Lemma 3.1, this gap directly affects the main theorem.","section":"§3, Lemma 3.3"},{"comment":"The uniqueness part of Theorem 1.1 is not proved in the same detail as the upper bound. For the case 3 | (n − 2), the argument is a single sentence asserting that the existence of a B3_5 creates a triangle inside it that does not intersect the 5-cycle containing u and v, and the cases 3 | (n − 1) and 3 | n are dismissed with 'by a similar discussion'. Since Theorem 1.1 explicitly claims the unique extremal graph for all three residue classes, the extremal characterization requires a written proof, including a verification of the graphs in Figure 9 for each residue class.","section":"§4, extremal characterization"}],"minor_comments":[{"comment":"The notation P_{ {n−2}/3 } is undefined; the fractional part of (n − 2)/3 is not a number of vertices. Please specify the final path as P_r for a small integer r, with the value of r determined by n mod 3.","section":"§1, Theorem 1.1"},{"comment":"The sentence 'B is a wheel {u} ∨ P_{t−1} or a fan {u} ∨ C_{t−1}' reverses the definitions from Section 2, where W_k = K1 ∨ C_{k−1} and F_k = K1 ∨ P_{k−1}; the text later also switches between u and v. Please correct the notation and the vertex names.","section":"§3, Lemma 3.5"},{"comment":"The proof states that any two blocks of B2_4 intersect in at most one vertex, and hence there are at most 10 such blocks, but this structural fact is not justified. Since Claim 3.2 uses the bound of 10, this step should be spelled out.","section":"§3, Claim 3.1"},{"comment":"There is a typo in 'all 3-faces in G incidenct with v'; it should read 'incident with v'.","section":"§4"},{"comment":"The inequalities in Claim 3.2 treat the number α of triangular blocks containing C5 as a real number; the floor and ceiling details in the counting argument should be clarified, especially since the final threshold n ≥ 295660 is derived from real-valued estimates.","section":"§3, Claim 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a meaningful problem and the claimed result is plausible, but the current manuscript is not in publishable form: the key structural lemmas, especially Lemma 3.4 and the classification inside Claim 3.2, are sketched rather than proved, and the extremal characterization is asserted rather than demonstrated. I see no indication of misconduct or circularity; the cited external bounds (Dowden's C5 result and triangular-block techniques) are used appropriately. The authors should be asked to provide complete proofs of these lemmas or to narrow the claims accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"New result: exact planar Turán number for C3∪C5, floor((8n-13)/3), with extremal characterization for n ≥ 295660. That fills a row in the table that previously only had pairs of 3- and 4-cycles. The lower-bound construction is clean, and the upper-bound strategy follows the triangular-block framework from Ghosh et al. The paper correctly uses Dowden's bound on C5-free planar graphs to force a C5 and then analyzes how triangular blocks interact with it. I don't see circularity; the two self-citations by P. Li are for adjacent results and the present proof doesn't reduce to them.\n\nThe soft spots are real. Lemma 3.4 is the hinge: it claims a triangular block with at least 521 vertices in a C3∪C5-free graph must be a wheel or fan. The low-degree case (Δ≤105) is plausibly repairable, but the high-degree case (Δ≥106) contains a chain of undefended assertions: that at most three fans can have ≥2 neighbors on a fixed 5-cycle C, that a fan with a single neighbor leads to an extension forcing a new vertex x, and that one of three edges must be in two 3-faces. The stress-test note is on target; the figures are illustrative, not a proof. If Lemma 3.4 fails, the upper bound collapses. Lemma 3.3 is also under-proved: the restriction of blocks to the six listed types relies on Lemma 3.2's enumeration of 6-vertex triangular blocks, and the completeness of that enumeration is asserted from diagrams. The extremal uniqueness for the non-residue cases is dismissed with 'by a similar discussion.'\n\nThat said, the gaps are not obviously fatal. The architecture is sound, and the numerical target is consistent with known bounds. A determined referee could likely fill the holes. This paper deserves a serious referee, not a desk reject. For anyone reading it, focus on Lemma 3.4; if that can be made rigorous, the rest likely follows.","headline":"Fills the C3∪C5 row in the planar Turán table, but the proof hinges on Lemma 3.4, which is under-proved.","tokens_in":7874,"tokens_out":4562,"would_cite":false,"duration_ms":41569,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C10","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every $n \\geq 295660$, the maximum number of edges in an $n$-vertex planar graph with no vertex-disjoint triangle and pentagon is $\\lfloor (8n-13)/3 \\rfloor$, attained by a unique extremal graph.","keywords":["planar Turán number","C3 ∪ C5","vertex-disjoint cycles","extremal graph","triangular block","forbidden subgraph","planar graph"],"falsifier":"Either of two observations would refute Theorem 1.1: an explicit $C_3 \\cup C_5$-free plane graph on $n \\geq 295660$ vertices with more than $\\lfloor (8n-13)/3 \\rfloor$ edges, or a triangular block with at least $521$ vertices that is neither a wheel nor a fan but is embeddable in a plane graph without creating a vertex-disjoint triangle and pentagon.","tokens_in":6829,"feed_emoji":"📐","tokens_out":16083,"duration_ms":149423,"temperature":0.7,"pith_summary":"The paper determines the exact planar Turán number of $C_3 \\cup C_5$: the largest number of edges in an $n$-vertex planar graph containing no vertex-disjoint triangle and pentagon. It proves that for every $n \\geq 295660$, this maximum is $\\lfloor (8n-13)/3 \\rfloor$, and that the unique extremal graph is $K_2$ joined to $\\lfloor (n-2)/3 \\rfloor$ copies of $P_3$ together with one short remainder path determined by $n$ modulo $3$. The result matters because exact values for planar Turán numbers of unions of two cycles were previously known only when both lengths were $3$ or $4$; the $(3,5)$ case adds a new exact value and a structural dichotomy that may transfer to other cycle pairs.","feed_headline":"Planar graphs without triangle-pentagon pair peak at (8n−13)/3 edges","feed_subtitle":"At n above 295,660, the densest graph is one edge plus many three-vertex paths.","key_machinery":"The proof is carried by the triangular block: a maximal union of $3$-faces glued edge-to-edge, into which every plane graph decomposes, with edges and $3$-faces summing over blocks. The pivotal mechanism is Lemma 3.4, which asserts that a triangular block with at least $521$ vertices in a $C_3 \\cup C_5$-free plane graph must be a wheel or a fan, so that all of its $3$-cycles pass through one vertex. In the complementary case, Lemma 3.1 forces many small triangular blocks containing pentagons through a fixed two-vertex set, and Lemma 3.3 restricts each such block to six small types satisfying $f_3(B_i) \\leq e(B_i)/2$; summing this over blocks and applying Euler's formula yields $e(G) \\leq (8n-13)/3$, with equality pinning down the extremal blocks that form the $K_2$-joined copies of $P_3$.","core_discovery":"The central claim is exact and structural. For every $n \\geq 295660$, $\\mathrm{ex}_{\\mathcal{P}}(n, C_3 \\cup C_5) = \\lfloor (8n-13)/3 \\rfloor$, and the only graph attaining it is $K_2$ joined to $\\lfloor (n-2)/3 \\rfloor$ disjoint copies of $P_3$ plus a path on the remaining $0$, $1$, or $2$ vertices. To prove this, the paper shows that any $C_3 \\cup C_5$-free plane graph with enough edges must fall into one of three configurations: many small triangular blocks each carrying a pentagon through a common two-vertex set, one triangular block of size at least $521$, or many blocks carrying pentagons through a single shared vertex. In the second and third configurations a structural lemma forces all $3$-cycles through one vertex, which Euler's formula shows is too sparse to reach the bound; in the first configuration the blocks are so restricted that the edge count cannot exceed the claimed value, and equality forces the $K_2$-to-$P_3$ construction.","pith_inferences":["The $295660$ threshold comes from the counting constants inside Lemma 3.1 and is almost certainly not sharp; the same formula may hold for much smaller $n$, and exact search for $n$ below the threshold could test this.","The $521$-vertex cutoff in Lemma 3.4 is likewise an artifact of the proof's counting rather than of the extremal phenomenon, so a sharper structural argument might lower it substantially.","The proof pattern (force a pentagon, decompose into triangular blocks, apply Euler's formula) is a plausible template for $C_3 \\cup C_k$ with larger odd $k$, though the extremal graph would likely replace the $P_3$ components by longer sparse pieces.","A natural next question is stability: whether planar graphs just below the extremal edge count must be close to the $K_2$-joined $P_3$ construction, and how many edges must be deleted from a maximal planar graph to kill every disjoint triangle-pentagon pair."],"forward_implications":["Any $n$-vertex planar graph with more than $\\lfloor (8n-13)/3 \\rfloor$ edges, for $n \\geq 295660$, must contain a triangle and a pentagon that are vertex-disjoint.","The unique extremal graph has a two-vertex cut: one shared edge joined to many disjoint three-vertex path components, with the remainder path determined by $n$ modulo $3$.","The extremal density is $8/3$ edges per vertex for all sufficiently large $n$, exactly $1/3$ below the trivial maximal planar density of $3$.","The theorem converts an avoidance problem into a sharp forcing statement: at that edge count, a disjoint triangle and pentagon are unavoidable in every planar graph."],"supporting_citations":[{"why":"Introduces triangular blocks and the decomposition into 3-face clusters that the proof uses throughout.","marker":"[6]"},{"why":"Supplies the planar Turán number of $C_5$, used to force the presence of a pentagon in any graph dense enough to matter.","marker":"[3]"},{"why":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, 2C_3)$, the triangle-pair case whose methods the present result extends.","marker":"[10]"},{"why":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, C_3 \\cup C_4)$, the immediate predecessor whose extremal structure guides this construction.","marker":"[12]"},{"why":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, 2C_4)$ and provides structural techniques for disjoint pairs of cycles.","marker":"[5]"},{"why":"Shows that three or more disjoint cycles are trivial, framing the two-cycle cases as the nontrivial range.","marker":"[9]"}],"fun_headline_variants":["Exact max edges for planar graphs avoiding C3∪C5: (8n−13)/3","Densest C3∪C5-free planar graph has (8n−13)/3 edges","Planar Turan number for C3∪C5: (8n−13)/3","Unique extremal planar graph for C3∪C5 has (8n−13)/3 edges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound rests on Lemma 3.4, which says that any triangular block with at least $521$ vertices in a $C_3 \\cup C_5$-free plane graph must be a wheel or a fan; if a large block can take another shape without producing the forbidden pair, the contradiction in the proof no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Exact max edges for planar graphs avoiding C3∪C5: (8n−13)/3","Densest C3∪C5-free planar graph has (8n−13)/3 edges","Planar Turan number for C3∪C5: (8n−13)/3","Unique extremal planar graph for C3∪C5 has (8n−13)/3 edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002046,"raw_usage":{"total_tokens":7982,"prompt_tokens":971,"completion_tokens":7011,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":6908}},"tokens_in":587,"tokens_out":7011,"duration_ms":49405,"temperature":1.0,"reasoning_tokens":6908,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:14:23.227044+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Either of two observations would refute Theorem 1.1: an explicit $C_3 \\cup C_5$-free plane graph on $n \\geq 295660$ vertices with more than $\\lfloor (8n-13)/3 \\rfloor$ edges, or a triangular block with at least $521$ vertices that is neither a wheel nor a fan but is embeddable in a plane graph without creating a vertex-disjoint triangle and pentagon.","supporting_citations":[{"cited_title":"Ghosh, E","cited_arxiv_id":null,"evidence_quote":"Introduces triangular blocks and the decomposition into 3-face clusters that the proof uses throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the planar Turán number of $C_5$, used to force the presence of a pentagon in any graph dense enough to matter."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, 2C_3)$, the triangle-pair case whose methods the present result extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, C_3 \\cup C_4)$, the immediate predecessor whose extremal structure guides this construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Determines $\\mathrm{ex}_{\\mathcal{P}}(n, 2C_4)$ and provides structural techniques for disjoint pairs of cycles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that three or more disjoint cycles are trivial, framing the two-cycle cases as the nontrivial range."}],"review_version":1}