{"id":"4f763342-34e7-4796-b0bb-fb11669433cf","arxiv_id":"2607.27166","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Claw-free graphs, already known to disprove the Stanley--Gasharov conjecture, are shown to yield infinitely many counterexamples in both line-graph and non-line-graph families, plus minimality of the base examples.","lead":"This paper constructs two infinite families of claw-free graphs whose chromatic symmetric functions have a negative Schur coefficient, contradicting the Stanley--Gasharov conjecture. It also certifies that two earlier small counterexamples are minimal under both vertex-first and edge-first size orderings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Q_t family rests on the unverified coefficient [s_(3,3,3,3,1)]X_Q = -144; a DP bug would invalidate the second infinite family and the Q minimality claims.","rationale":"The reader's verdict is CONDITIONAL, with the weakest assumption identified as computational correctness of the exhaustive searches and the DP, plus the unverified Q coefficient. I agree: the mathematical core—the transfer lemma (Lemma 3.3) and the claw-free/non-line-graph structural arguments—is sound. The F_t family depends only on the externally reproduced G2 coefficient, so it is essentially immune to the paper's own computational vulnerabilities. The Q_t family, however, depends entirely on Eq. (4.3), the value -144 for a 13-vertex graph computed by the authors' own DP. The DP itself is described in enough detail to be reimplemented, and the paper mentions internal cross-checks, but no code or data is shipped. A bug in the DP would directly falsify the second infinite family and the stronger minimality claims for non-line-graph counterexamples. The reader's weakest_assumption already names this (the Q coefficient) as a concern, so our read agrees. I do not see an internal mathematical inconsistency in the transfer arguments or the graph constructions; the only risk is empirical/reproducibility. Therefore the appropriate verdict remains CONDITIONAL: accept the mathematical reduction but require independent verification of the key computation before full confidence. The editorial mislabeling of Conjecture 1.1 as 'Lemma 1.1' is irrelevant to correctness.","tokens_in":10430,"tokens_out":30351,"duration_ms":286067,"concrete_test":"Independently compute the complete Schur expansion of X_Q using a different implementation (e.g., SageMath's symmetric-function package, or a separately written DP following the §2.2 specification) and verify that the coefficient of s_(3,3,3,3,1) is exactly -144. As a secondary check, recompute the Schur expansion of F3 (which the paper states is Schur-positive) with the same independent tool and confirm no negative coefficients, since that would catch systematic sign errors in the DP. If available, also compare a handful of random graphs from the §2.3 census against the same independent implementation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The infinite family Q_t (Theorem 4.4) is obtained by applying the transfer lemma (Lemma 3.3) to the base graph Q. The transfer lemma is proved rigorously, so the existence of the Q_t family is logically equivalent to the truth of Eq. (4.3): [s_(3,3,3,3,1)]X_Q = -144. This coefficient is asserted from a 'direct exact computation' using the frontier-connectivity DP of §2.2, but no code, data, or independent verification is supplied. The DP is described in detail and appears correct, but a single implementation bug would change the sign or magnitude of Eq. (4.3) and would collapse the entire second family. In contrast, the F_t family rests on the G2 coefficient from two independent external sources (Prajapati; Matherne–Morales), so it is much less exposed. The minimality results (Propositions 2.1 and 4.2) additionally depend on the same DP and on Prajapati's external census, but even if those censuses were wrong, the paper would still have one infinite family; only the Q_t family and the non-line-graph minimality would fail if Eq. (4.3) is wrong. Thus the single most load-bearing unverified premise is the computed Schur coefficient of Q.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Stanley–Gasharov conjecture that every claw-free graph is Schur-positive, which was recently disproved by two independent pairs of authors. The manuscript constructs two infinite families of counterexamples. The first family consists of line graphs F_t (t≥4), obtained by adjoining K_t to the 12-vertex counterexample G_2 through a single bridge; the paper proves the Schur coefficient of (4,4,4,4,1^{t-4}) in X_{F_t} is −40(t−1)(t−1)!. The second family consists of non-line-graphs Q_t (t≥5), obtained from a new 13-vertex claw-free non-line-graph Q by the same bridge construction, with coefficient −144(t−1)(t−1)! of (4,4,4,4,2,1^{t-5}). The paper also reports exhaustive computations establishing that G_2 is the edge-first minimum counterexample and that Q is the minimum non-line-graph counterexample under both vertex-first and edge-first orders. The main theoretical tool is a transfer lemma (Lemma 3.3) showing that the bridge-plus-clique operation multiplies a chosen Schur coefficient by (t−1)(t−1)! and shifts the partition by a full column.","tokens_in":10749,"tokens_out":20641,"duration_ms":198949,"significance":"If the computational assertions are correct, the paper answers an explicit question of Matherne and Morales and gives the first infinite families of counterexamples, including a non-line-graph family. The transfer lemma is elementary, parameter-free, and rigorously proved; it is a clean contribution that may be useful beyond this example. The first family is robust because its base coefficient for G_2 comes from two independent external sources. The second family, however, rests entirely on the computed coefficient (4.3) for Q, for which no code, data, or independent verification is supplied; this is the main risk. The reported exhaustive censuses are extensive and internally consistent, but not independently auditable from the manuscript. Should the code and certificates be made available, the paper would be a solid and significant contribution. The paper is honest about the computational nature of several proofs and provides useful structural context via quasi-line graphs and Chudnovsky–Seymour theory.","major_comments":[{"comment":"The coefficient [s_(3,3,3,3,1)]X_Q = −144 is asserted from 'a direct exact computation' using the frontier-connectivity DP of §2.2, but no code, data, or independent confirmation is provided. This value is the sole base input for the second infinite family Q_t (Theorem 4.4, Eq. (4.6)) and for the non-line-graph minimality claims. If this coefficient is incorrect, the entire Q_t family and the non-line-graph minimality results collapse. Please make the computation checkable by supplying the program, the power-sum expansion of X_Q, and the final Schur-coefficient table, or provide an independent verification by a different method.","section":"§4.1, Eq. (4.3)"},{"comment":"The exhaustiveness claims for the edge-first and vertex-first censuses are not auditable. The paper states that geng generated 144,492 graphs in §2 and 16,087,992 in §4, and that a DP and a line-graph filter were applied, but no code, logs, hashes, or output files are included. Propositions 2.1 and 4.2 depend on the correctness and completeness of these computations, as does the uniqueness of Q. Please archive the full computational pipeline (generator inputs, filters, and result files) or provide a certificate, such as the list of all found negative coefficients and the exact commands used to produce each row of Tables 1–3.","section":"§2.3 and §4.1, Tables 1–3"}],"minor_comments":[{"comment":"The conjecture is consistently called 'Lemma 1.1' after the introduction (e.g., §1, §2, Proposition 2.1, Theorem 3.4). This should be 'Conjecture 1.1'.","section":"Throughout"},{"comment":"The proof references 'Lemma 3.1' and 'Lemma 3.4', but the intended statements are Proposition 3.1 and Theorem 3.4. Fix the cross-references.","section":"§3, proof of Theorem 3.4"},{"comment":"The sentence 'each non-claw obstruction has at least seven edges' is terse and unsupported. Since this bound helps define the search range, please give a reference to Beineke's list or a short justification for why every claw-free non-line-graph obstruction has at least seven edges.","section":"§4.1, Eq. (4.4)"},{"comment":"The row 'n=13, 21≤m≤27' in Table 3 overlaps the target stratum in Table 2. A sentence clarifying which computations were rerun and how the overlap was handled would improve reproducibility.","section":"Tables 2–3"},{"comment":"The two arXiv preprints [8] and [11] are cited without version identifiers or DOIs. Please add the full version numbers, as the accompanying data is versioned.","section":"References [8], [11]"}],"recommendation":"major_revision","confidential_remarks":"The first infinite family is convincing and rests on independently verified base data. The main issue is reproducibility of the Q coefficient and of the large censuses; I would be comfortable with acceptance once the authors provide code and data or an independent verification. The paper is in scope for a combinatorics journal and the transfer lemma is a genuinely useful tool."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely reusable piece here is Lemma 3.3, the Schur-coefficient transfer lemma: adjoining K_t through a single bridge multiplies a specified Schur coefficient by (t−1)(t−1)!. The proof is short, rigorous, and does exactly what you want a lemma to do. The F_t family follows immediately from the known G2 coefficient −40, which is backed by two independent external sources (Prajapati; Matherne and Morales), so that half of the paper is on solid ground. I also think the edge-first minimality result for G2, though computational, is a legitimate answer to a natural gap in the existing census.\n\nThe soft spot is exactly where the stress-test note lands. The Q_t family is logically equivalent to the asserted coefficient [s_(3,3,3,3,1)]X_Q = −144, and that number comes from the paper's own frontier-connectivity DP, with no shipped code or data. A single bug in that DP would collapse the second infinite family and the Q minimality claims. The description of the DP is detailed and the independent checks they mention (recounting generator scopes, comparing the line-graph filter) are reassuring, but \"described in enough detail to reimplement\" is not the same as \"independently auditable.\" The same issue colors the big census claims in Proposition 4.2: they depend on geng runs and on Prajapati's external census files, which are not bundled here. I would not call this a mathematical flaw—the reasoning is clear—but it is a reproducibility gap. The repeated \"Lemma 1.1\" for the Stanley–Gasharov conjecture is an editorial irritant, nothing more.\n\nIf I were refereeing, I would send it out and ask for the computation artifacts as a condition of acceptance. The core transfer lemma is certainly correct, and one infinite family is unambiguous even if the second family's computational premise were somehow wrong. That is enough to justify referee time, but the authors should be pushed to ship the verification code and the census data rather than leave the load-bearing coefficient as an act of faith.","headline":"The transfer lemma is clean and gives one solid infinite family from the known G2 counterexample; the second family and minimality claims hinge on a single unshipped DP computation, so the referee should ask for code or independent verification.","tokens_in":11149,"tokens_out":1310,"would_cite":true,"duration_ms":17695,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C15","05C76"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Stanley–Gasharov conjecture is false for two infinite families of claw-free graphs, one of line graphs and one of non-line graphs, with explicit negative Schur coefficients.","keywords":["chromatic symmetric function","claw-free graph","line graph","Schur positivity","Stanley–Gasharov conjecture","Schur-coefficient transfer","quasi-line graph"],"falsifier":"Recompute the Schur expansion of F_4 by an independent implementation and check that the coefficient of s_{(4,4,4,4)} equals −120; or rerun an independent enumeration of connected claw-free graphs on 13 to 21 vertices with 12 to 20 edges and check that none has a negative Schur coefficient.","tokens_in":10359,"feed_emoji":"","tokens_out":5263,"duration_ms":50568,"temperature":0.7,"pith_summary":"The paper establishes that the Stanley–Gasharov conjecture, which predicted Schur-positivity for every claw-free graph, fails not just for a few exceptional graphs but for two entire infinite families. The first family is made of line graphs; the second is made of graphs that are not line graphs, so no refinement to the line-graph class rescues the conjecture. The engine is a transfer rule: attaching a large clique to a vertex through a single edge multiplies a chosen Schur coefficient by (t−1)(t−1)! and shifts the indexing partition by one full column, turning one negative coefficient into infinitely many. Along the way the paper verifies, by exhaustive exact computation, that the base counterexamples are the unique smallest ones under two natural vertex/edge orderings.","feed_headline":"Two infinite families break the Stanley–Gasharov conjecture","feed_subtitle":"The conjecture said every claw-free graph has Schur-positive coloring; line graphs and non-line graphs now fail it.","key_machinery":"The Schur-coefficient transfer lemma (Lemma 3.3). It states that for any graph G, vertex v, and t≥1, the chromatic symmetric function of the graph B_t(G,v) obtained by joining a disjoint clique K_t to G by the single edge v–w satisfies ρ_t(X_{B_t(G,v)}) = (t−1)(t−1)! e_t(x_1,…,x_t) ρ_t(X_G), where ρ_t sets all but t variables to zero. Combined with the identity e_t s_λ = s_{λ+(1^t)} for partitions with at most t parts, this transfers a Schur coefficient of X_G to a Schur coefficient of X_{B_t(G,v)} with the same sign, scaled by (t−1)(t−1)!. This is the mechanism that turns the single counterexamples G_2 and Q into infinite ladders of counterexamples.","core_discovery":"For every t≥4, the line graph F_t obtained by joining G_2 to a clique K_t by one edge satisfies ⟨s_{(4,4,4,4,1^{t-4})}, X_{F_t}⟩ = −40(t−1)(t−1)!, and for every t≥5, the non-line-graph Q_t obtained from Q by the same construction satisfies ⟨s_{(4,4,4,4,2,1^{t-5})}, X_{Q_t}⟩ = −144(t−1)(t−1)!. Since these coefficients are strictly negative, both families violate the conjecture. The construction is uniform: Lemma 3.3 shows that adjoining K_t to any graph at a vertex v multiplies the Schur coefficient indexed by λ by (t−1)(t−1)! and replaces λ by λ+(1^t), provided ℓ(λ)≤t. Applied to the base counterexamples G_2 (with negative coefficient −40) and Q (with −144), this yields the infinite families","pith_inferences":["The transfer lemma is general: any graph with a negative Schur coefficient indexed by a partition with at most t parts yields an infinite family of larger graphs with negative coefficients. This suggests counterexamples to Schur-positivity are abundant and can be generated from a single seed by a uniform operation.","The construction preserves claw-freeness only for carefully chosen attachment vertices (here v_3 in G_2 and u_1 in Q). It would be interesting to test whether the same amplification works for other claw-free base graphs with negative coefficients, or whether the choice of attachment vertex is forced by the claw-free condition.","The exhaustive search boundaries suggest a threshold phenomenon: no counterexamples exist below sizes 12/21 or 13/27, but infinitely many exist above. A natural next question is whether every claw-free graph above some other threshold is non-Schur-positive, or whether counterexamples remain sparse in some density sense."],"forward_implications":["The Stanley–Gasharov conjecture is false as originally stated, and the failure is robust: there are infinitely many claw-free counterexamples, including line graphs and non-line graphs.","The transfer lemma provides a general method to build new counterexamples from any graph with a negative Schur coefficient, so the phenomenon is not confined to the two base graphs.","The minimality results give a complete picture of the small end: every connected claw-free graph with at most 11 vertices, or with 12 vertices and at most 20 edges, is Schur-positive; G_2 is the unique 12-vertex counterexample and Q the unique non-line-graph counterexample with 13 vertices.","The non-line-graph family shows that even quasi-line graphs—the next structural class above line graphs—contain counterexamples, so the boundary of the conjecture lies strictly above line graphs."],"fun_headline_variants":["Two infinite families refute Stanley–Gasharov","Line and non-line graphs yield infinite counterexamples","Infinite counterexamples: line graphs and non-line graphs","Stanley–Gasharov falls to two infinite counterexample families"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the exhaustive graph generation and the exact Schur-coefficient computations are complete and bug-free; if the searches missed any connected claw-free graph in the stated ranges, or if a negative coefficient were miscalculated, the minimality claims would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Two infinite families refute Stanley–Gasharov","Line and non-line graphs yield infinite counterexamples","Infinite counterexamples: line graphs and non-line graphs","Stanley–Gasharov falls to two infinite counterexample families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000711,"raw_usage":{"total_tokens":3095,"prompt_tokens":861,"completion_tokens":2234,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":2166}},"tokens_in":605,"tokens_out":2234,"duration_ms":19274,"temperature":1.0,"reasoning_tokens":2166,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:43:12.418738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the Schur expansion of F_4 by an independent implementation and check that the coefficient of s_{(4,4,4,4)} equals −120; or rerun an independent enumeration of connected claw-free graphs on 13 to 21 vertices with 12 to 20 edges and check that none has a negative Schur coefficient.","supporting_citations":[],"review_version":2}