{"id":"dee11f85-bb45-4f77-8363-880694846f48","arxiv_id":"2607.21508","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Two line graphs with 12 vertices have non-Schur-positive chromatic symmetric functions, and a bipartite Schur-positive graph has an unsaturated Newton polytope, disproving two conjectures.","lead":"This paper presents explicit 12-vertex graphs that refute Stanley's 1995 conjecture that chromatic symmetric functions of claw-free graphs expand positively in Schur functions. It also gives a bipartite Schur-positive graph whose Newton polytope is not saturated, disproving a 2018 conjecture by Monical.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"G1/G2 counterexamples to Stanley's conjecture rest on unshown SageMath coefficients; a transcription/computation error would invalidate them, so independent recomputation is needed.","rationale":"The mathematical structure is sound: the G3 proof is rigorous, the line graphs are claw-free, and edge counts match. The only serious risk is the unshown computation for G1/G2. I checked the non-saturation argument: the counting of non-neighbors is slightly terse but correct in context, and the K_{2,2} argument rules out weight (7,4,1). The displayed G3 Schur expansion is extensive and all-positive, though the 'XG2' label appears to be a typo for 'XG3'; its correctness is still computational. Therefore the reader's CONDITIONAL verdict remains right: the paper's central claim is plausible but not fully independently verifiable from the text alone. An independent recomputation of the s_3333 coefficients would settle it.","tokens_in":6575,"tokens_out":14800,"duration_ms":133842,"concrete_test":"Recompute the full Schur expansion of X_G for the graphs H1 and H2 given in Examples 2.1 and 2.2 using an independent implementation (e.g., SageMath's chromatic_symmetric_function followed by schur, or a proper-coloring DP computing Schur coefficients); check that the coefficient of s_3333 equals -64 for G1 and -40 for G2. If either coefficient differs, the counterexample to Stanley's conjecture fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central disproof of Conjecture 1.1 is a computational assertion: for the two explicit line graphs G1 and G2, the coefficient of s_3333 in X_G is reported as -64 and -40, with only 'see code' and no code or full Schur expansion included. Because the graph data in Examples 2.1/2.2 is explicit, this is checkable, but if the SageMath script or edge-list transcription is wrong, the negative coefficients are not established. The G3 SNP counterexample is different: the non-saturation argument is rigorous and self-contained, and a full positive Schur expansion for G3 is displayed (despite the typo 'XG2' in its label). Thus the load-bearing gap is specifically the unverified computational coefficients for G1/G2 (and, secondarily, the accuracy of the displayed G3 expansion).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript gives two explicit 12-vertex line graphs G1 and G2, with edge lists in Examples 2.1 and 2.2, for which it claims the Schur coefficient [s_3333]X_G is -64 and -40, respectively. Since line graphs are claw-free, this would disprove Stanley's 1995 conjecture that claw-free graphs have Schur-positive chromatic symmetric functions. It also gives a 12-vertex bipartite graph G3, with edge list in Example 3.1, for which it claims X_G3 is Schur-positive but the three-variable Newton polytope is not saturated: the monomials x_6^6 x_6^6 and x_8^8 x_2^2 x_2^2 occur while their average x_7^7 x_4^4 x_1 does not, contradicting Monical's 2018 SNP conjecture for Schur-positive chromatic symmetric functions. The non-saturation argument for G3 is elementary and self-contained; the Schur positivity of G3 and the negative coefficients for G1/G2 are presented as SageMath computations.","tokens_in":6811,"tokens_out":11908,"duration_ms":113417,"significance":"If the computational claims are correct, the paper settles two longstanding conjectures in the negative with explicit, small, checkable graphs. The G3 non-saturation proof is elegant and can be verified by hand from the displayed edge list. The paper also situates the examples in the context of recent work by the authors and by Prajapati, and the explicit data makes independent recomputation feasible. The main weakness is that the decisive computations are only referenced as 'see code' and are not included in the manuscript, so the central disproofs are not yet reproducible from the text.","major_comments":[{"comment":"The claims [s_3333]X_G1=-64 and [s_3333]X_G2=-40 are the central disproof of Conjecture 1.1. The text says only 'Using SageMath (see code)' but no code, output, or certificate is provided. Since the entire counterexample rests on these two coefficients, the manuscript must include the complete SageMath script or a full Schur expansion (or another independently checkable certificate) so that a reader can verify the negative coefficients without reimplementing the computation from scratch.","section":"Section 2, Examples 2.1 and 2.2"},{"comment":"The Schur positivity of X_G3 is load-bearing for the counterexample to Monical's conjecture, but the displayed expansion is not reproducible as written. The notation is undefined and appears corrupted: e.g. 's_112', 's_2110', 's_32215' are not standard partition labels, and the expansion is labeled 'XG2' instead of 'XG3'. The paper needs to give the actual SageMath code that produces the expansion, or a machine-readable version in a clearly defined notation, so that the positivity claim can be checked.","section":"Section 3, displayed Schur expansion of G3"},{"comment":"The promised computational artifacts are not present. The references 'see code' and 'see example of the output here' are dangling; no code, output, or prompts are included. This is a presentation issue for Section 4, but it directly affects the verifiability of the main results because those artifacts are the only source for the G1/G2 Schur coefficients and the G3 Schur expansion.","section":"Section 4"}],"minor_comments":[{"comment":"The label 'XG2' in the displayed Schur expansion should be 'XG3'.","section":"Section 3"},{"comment":"The argument is correct but terse. Expanding the sentence about the vertices 1,2,3,4 and the remaining outer vertex into an explicit case analysis would improve readability.","section":"Section 3, proof of no weight (7,4,1)"},{"comment":"The shorthand for Schur function indices (e.g. 's_112', 's_2110') should be defined, or standard partition notation with commas/exponents should be used.","section":"Section 3, notation"},{"comment":"The phrase 'Both of these examples' is slightly misleading since the paper contains three graphs (G1 and G2 for Stanley's conjecture, G3 for Monical's). It would be clearer to say 'Both counterexamples' or 'The counterexamples'.","section":"Abstract and Introduction"},{"comment":"The hyperlinks 'here' are placeholders. Either include the actual prompt/output files or remove them; in the current form the section promises material that is not part of the manuscript.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The mathematical idea is strong, the graphs are explicit, and the G3 non-saturation proof is rigorous and elegant. The sole substantive obstacle is the missing computational evidence. If the authors include the SageMath code or a verifiable full expansion, this should be acceptable. Separately, the prominent disclosure that the examples were found using ChatGPT-5.6 Sol Pro is unusual; the editors may want to check the journal's policy on AI-assisted discovery and on the inclusion of such statements in the main text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this note is real: it gives explicit 12-vertex line graphs whose chromatic symmetric functions have negative Schur coefficients, refuting Stanley's 1995 claw-free conjecture, and a bipartite graph with Schur-positive X but unsaturated Newton polytope, refuting Monical's conjecture. Second, the two disproofs of Stanley's conjecture rest on SageMath outputs that are only referenced as 'see code'—the paper displays negative coefficients (-64 and -40) but no code or expansion. The graph data is explicit, so the claim is checkable, but the manuscript alone does not establish it.\n\nThe best part of the paper is the Monical counterexample. The G3 SNP argument is elegant and self-contained: weights (6,6,0) and (8,2,2) occur, their average (7,4,1) does not, with a short counting proof. A full (if typo-labeled) Schur expansion is displayed, so the Schur-positivity claim is at least given in the text. The paper also treats the literature honestly: it credits Prajapati's independent simultaneous counterexample and Wang–Zhang–Zhao's infinite family, and notes that its own examples were found via ChatGPT. That last detail is unusual but the math stands independently of the tool.\n\nWhere it is soft: the two line-graph counterexamples are load-bearing computational assertions. If the SageMath script or the edge-list transcription is off, the negative coefficients are not established. This is a fixable gap—ancillary code or a printed Schur expansion would close it—but as submitted, the paper's main claim is conditional on external artifacts. The G3 expansion also has a typo ('XG2' in the label) and no code; it is less concerning because the SNP counterexample does not depend on it. For a counterexample claim, I would want the verification to be entirely in the record. The 'related work' section is well handled; the citation pattern looks appropriate.\n\nBottom line: this paper is for algebraic combinatorialists who care about Schur positivity, Newton polytopes, and the status of old conjectures. It deserves a serious referee, and the answer should be 'accept after the authors include code or full expansions'. The mathematical architecture is sound; the missing evidence is a verification artifact, not a sign of a flawed idea. If the computations check out, this is a clean refutation of two conjectures and a good research note.","headline":"A short, important note refuting Stanley's claw-free Schur positivity conjecture with explicit line graphs; the non-Schur-positive coefficients are computational and need to be made checkable.","tokens_in":7218,"tokens_out":2553,"would_cite":true,"duration_ms":27268,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05C15","05C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two claw-free graphs disprove a 1995 Schur-positivity conjecture","keywords":["chromatic symmetric function","Schur positivity","claw-free graph","line graph","Newton polytope","saturated Newton polytope","counterexample","graph coloring"],"falsifier":"Independently recompute the Schur expansion of X_G for the two 12-vertex line graphs from the displayed edge lists and check whether the coefficient of s_3333 is -64 and -40; also compute the full Schur expansion of X_{G3} and exhaustively test all 3-colorings of G3 for a proper coloring with color-class sizes (7,4,1). A positive s_3333 coefficient, a nonzero m_741 coefficient, or any negative Schur coefficient for G3 would refute the corresponding claim.","tokens_in":6494,"feed_emoji":"🎨","tokens_out":5280,"duration_ms":57320,"temperature":0.7,"pith_summary":"A long-standing conjecture asserted that every claw-free graph has a chromatic symmetric function with nonnegative Schur coefficients. This paper exhibits two 12-vertex line graphs—automatically claw-free—whose chromatic symmetric functions have negative s_3333 coefficients, -64 and -40. It also gives a 12-vertex bipartite graph whose chromatic symmetric function is Schur-positive yet whose Newton polytope in three variables is not saturated: the color-weight vectors (6,6,0) and (8,2,2) occur, but their average (7,4,1) does not. If the computations are correct, both the 1995 claw-free conjecture and the 2018 saturation conjecture are false as stated.","feed_headline":"Twelve-vertex line graphs break Schur-positivity","feed_subtitle":"Explicit counterexamples also sink the saturation conjecture for chromatic symmetric functions.","key_machinery":"The chromatic symmetric function X_G(x) = sum over proper colorings κ of x_{κ(1)}...x_{κ(n)}, together with its expansion in the Schur symmetric function basis; a function is Schur-positive exactly when every Schur coefficient is nonnegative. The line graph operation L(H)—vertices are edges of H, adjacency means sharing a vertex—is the device that guarantees claw-freeness while producing graphs with nontrivial Schur behavior. For the saturation counterexample, the Newton polytope of X_G(x1,x2,x3) serves as the key object: its lattice points are the color-weight vectors arising from proper colorings, and the paper shows that two such vectors occur while their lattice average does not.","core_discovery":"The paper centers on the chromatic symmetric function X_G, which records a monomial for every proper coloring of G, and on its expansion in the Schur basis. The authors produce two line graphs G1 and G2, obtained from 10-vertex graphs with 12 edges each, for which the s_3333 Schur coefficient of X_G is -64 and -40, respectively. Since every line graph is claw-free, these are explicit counterexamples to the claim that all claw-free graphs are Schur-positive. Independently, the authors give a 12-vertex bipartite graph G3 whose chromatic symmetric function is Schur-positive—they display the full expansion—but whose Newton polytope in three variables fails to be saturated: proper colorings with","pith_inferences":["Editorial extension: the non-saturation of G3 at k=3 suggests that the SNP property may fail for many Schur-positive chromatic symmetric functions; a natural test is whether other small bipartite graphs with Schur-positive X_G also have unsaturated Newton polytopes.","Editorial extension: the minimality claim for G2 could be independently verified by exhaustive enumeration of all claw-free graphs below that size; any new small graph with a negative Schur coefficient would either confirm the reported minimality or refine it.","Editorial extension: the method of searching by computer for negative Schur coefficients points toward a feasible automated program: enumerate claw-free graphs by edge count, compute a finite window of Schur coefficients, and stop at the first negative one. This could reveal infinite families rather than isolated examples."],"forward_implications":["The 1995 conjecture that claw-free graphs are Schur-positive is false; claw-freeness alone does not force Schur positivity.","The 2018 conjecture that every Schur-positive chromatic symmetric function has saturated Newton polytope in all numbers of variables is false; the failure occurs already in three variables.","The 12-vertex threshold for the saturation conjecture is sharp, matching the prior lower bound from earlier work.","The two line-graph counterexamples lie outside the previously known Schur-positive class of claw-free incomparability graphs, so positive results for that class do not extend to all claw-free graphs.","The explicit edge lists give concrete test graphs for any future conjecture about claw-free Schur positivity or Newton-polytope saturation."],"fun_headline_variants":["Line graphs crush Schur positivity","12-vertex line graphs disprove Stanley's conjecture","Negative Schur coefficients in claw-free graphs","Saturation conjecture fails for Schur-positive graphs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The counterexamples to the first conjecture rest on computer calculations of Schur expansions that are cited but not printed in the paper; if the code or the transcription of the edge lists is wrong, the negative coefficients -64 and -40 would not be established, and the Schur-positivity of G3 also depends on an unshown machine expansion.","fun_headline_variants_meta":{"raw":{"variants":["Line graphs crush Schur positivity","12-vertex line graphs disprove Stanley's conjecture","Negative Schur coefficients in claw-free graphs","Saturation conjecture fails for Schur-positive graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1603,"prompt_tokens":630,"completion_tokens":973,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":374,"completion_tokens_details":{"reasoning_tokens":917}},"tokens_in":374,"tokens_out":973,"duration_ms":11278,"temperature":1.0,"reasoning_tokens":917,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:48:08.559385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently recompute the Schur expansion of X_G for the two 12-vertex line graphs from the displayed edge lists and check whether the coefficient of s_3333 is -64 and -40; also compute the full Schur expansion of X_{G3} and exhaustively test all 3-colorings of G3 for a proper coloring with color-class sizes (7,4,1). A positive s_3333 coefficient, a nonzero m_741 coefficient, or any negative Schur coefficient for G3 would refute the corresponding claim.","supporting_citations":[],"review_version":2}