{"id":"0786a74d-4840-4d23-a442-2df937fa813e","arxiv_id":"2508.13643","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For C_{2k+1}-free graphs with chromatic number at least r, the authors prove stronger structural stability and the first spectral extremal result, with unique extremal graph T_{n-r+1,2} ∘ K_r.","lead":"This paper proves spectral and structural stability results for graphs with no odd cycle of length 2k+1, showing such graphs are nearly bipartite unless they are small. It also gives the first spectral extremal solution for these graphs under a high chromatic number constraint, naming the exact extremal graph.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's structural decomposition is too coarse; the spectral theorem's unique equality case requires an exact K_r-attachment lemma that is not stated, so the central claim is uncheckable from the available text.","rationale":"The reader's weakest-assumption identification was the structural stability theorem, and I agree that it is the load-bearing premise. However, I sharpen the concern: the abstract's structural statement is not just unproven but too weak to logically force the unique spectral extremal graph. The equality case T_{n-r+1,2}∘K_r requires exact control of how the suspended vertices are connected to each other and to the bipartite core, not merely a bound on their number. Without that control, spectral radius could plausibly be larger for a different suspension pattern, so the theorem's correctness depends on an unstated exact-stability lemma. This is a genuine risk for the central claim. I am not recommending rejection because the full text may well contain the missing lemma, and the result could be correct. The appropriate action is to keep the UNVERDICTED verdict pending inspection of the full proof. My concern is internal to the argument's completeness, not a challenge based on external consensus, and I hold the authors in good faith as having possibly supplied the missing details in the full manuscript.","tokens_in":1155,"tokens_out":10020,"duration_ms":112150,"concrete_test":"In the full text, locate the lemma in the spectral part that derives λ(G) ≤ λ(T_{n-r+1,2}∘K_r). Verify whether it proves the suspended part is exactly a K_r attached to one vertex of the smaller partite set. If it only uses the coarse r-2 deletion decomposition, run a targeted check for small parameters: for r=3, k=2, n=20, generate all (or sample via SDP) C5-free graphs with chromatic number at least 3 and check whether the spectral-radius maximum is uniquely attained by T_{n-2,2}∘K_3. If a graph with larger λ exists or the maximizer is non-unique, the theorem's statement fails; if the maximizer is unique, the coarse decomposition may be sufficient, but the missing lemma must still be explicit in the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The spectral theorem asserts a unique extremal graph: T_{n-r+1,2}∘K_r, in which a single K_r is suspended from one vertex of the smaller partite set of the bipartite Turán graph. For this equality claim to hold, the proof must show that in any extremal C_{2k+1}-free graph with χ≥r, the non-bipartite part is exactly this K_r attached in this specific way. The structural theorem as summarized in the abstract only says G is obtained from a large bipartite graph by 'suspending some small graphs' with total vertex count at most r-2. It does not constrain (a) the internal edges among the suspended vertices, or (b) whether the suspension attaches to one vertex of the smaller part or spreads across both parts. Spectral radius is not a function of edge count alone: changing the attachment pattern or adding one edge inside the suspended set changes λ without changing the r-2 deletion bound. Thus the eigenvalue inequality λ(G) ≤ λ(T_{n-r+1,2}∘K_r) does not follow from the coarse deletion statement alone; some additional exact-stability lemma must be doing the work. That lemma is part of the paper's central claim but is absent from the abstract, so the reader cannot verify whether the step from 'made bipartite by deleting r-2 vertices' to 'extremal graph is the Turán-plus-clique join' is rigorous. This is an internal-correctness risk, not a disagreement with prior results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims two main results. First, for C_{2k+1}-free graphs on n vertices with at least the bipartite Turán number plus a clique term, it establishes a structural stability theorem: such graphs can be made bipartite by deleting at most r-2 vertices, and more strongly, they are obtained from a large bipartite graph by suspending small graphs whose total order is at most r-2. This improves prior bounds on n by Ren-Wang-Wang-Yang and by Yan-Peng. Second, for the spectral extremal problem, the paper claims that for 3≤r≤2k and n≥712k, every n-vertex C_{2k+1}-free graph with chromatic number at least r has spectral radius at most λ(T_{n-r+1,2}∘K_r), with equality only for the stated graph. The authors state this is the first solution to the spectral extremal problem for F-free graphs with high chromatic number.","tokens_in":1474,"tokens_out":2100,"duration_ms":23871,"significance":"If the proofs are correct, the spectral theorem is a significant advance: it unifies and extends the prior results of Guo-Lin-Zhao and Zhang-Zhao, and it gives the first spectral extremal result for F-free graphs with a chromatic number lower bound. The structural theorem also improves quantitative dependencies, replacing quadratic bounds with a linear bound (n≥712k for the spectral part), which is valuable. The paper has no fitted constants, and the equality cases are explicitly stated, both of which are strengths.","major_comments":[{"comment":"The structural theorem is stated only as 'roughly' describing G as a large bipartite graph plus 'some small graphs' with total vertex count at most r-2. This formulation is too weak to support the spectral theorem's unique equality case. The equality G = T_{n-r+1,2}∘K_r requires that the suspended part is exactly one K_r attached to a vertex of the smaller partite set, with no additional edges among suspended vertices and no other attachment pattern. Merely deleting r-2 vertices to make G bipartite does not determine the spectral radius, since different internal edges and attachment positions change λ without changing the deletion count. The abstract needs to state the exact decomposition lemma, or at least a precise version that constrains the suspended subgraph and its attachment, for the implication to be checkable.","section":"Abstract, first paragraph"},{"comment":"The proof of the spectral theorem is not described. The statement asserts an extremal graph with a unique equality case, but the abstract provides no indication of the argument connecting the structural decomposition to the spectral radius comparison. As written, the reader cannot verify whether the step from 'bipartite after deleting r-2 vertices' to 'the unique extremal graph is T_{n-r+1,2}∘K_r' is rigorous. This is a load-bearing gap in the summary. The full text may contain the required argument, but the abstract does not; since no full text is available for this review, the central claim remains uncheckable.","section":"Abstract, second paragraph"}],"minor_comments":[{"comment":"The phrase 'suspending some small graphs' is informal; the paper should specify whether these graphs are cliques, independent sets, or arbitrary, and how they are attached to the bipartite core.","section":"Abstract, first paragraph"},{"comment":"The definition of T_{n-r+1,2}∘K_r says 'identifying a vertex of K_r and a vertex of the smaller partite set' but does not say which vertex of K_r; this should be made precise (the paper presumably specifies any vertex, but the abstract leaves it ambiguous).","section":"Abstract, second paragraph"},{"comment":"The improvement statement 'weakening the requirement on n and k' is vague; the prior results' hypotheses should be compared explicitly in the introduction/theorems.","section":"Abstract, first paragraph"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the arXiv abstract, as no full text was provided. The stress-test concern about the gap between the coarse structural statement and the unique spectral equality case is legitimate on the face of the abstract. However, this is likely an artifact of abstract-level exposition rather than a definite error in the full paper. Because the central claims are proof-based and no derivations are available, I cannot verify soundness. I recommend requesting the full manuscript before making a substantive decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is an abstract-only submission, so no proof can be checked. If the theorems are correct, this is a genuine step forward: it improves the stability threshold for C_{2k+1}-free graphs with chromatic number at least r from quadratic to linear in n, strengthens the structural conclusion, and gives the first spectral extremal result for F-free graphs with high chromatic number. The extremal graph, T_{n-r+1,2}∘K_r, is the obvious candidate, so the novelty is in the proof technology.\n\nThe main thing I'd want to see before trusting the spectral theorem is the lemma that bridges the structural statement to the sharp equality case. The abstract only says the graph becomes bipartite after deleting at most r-2 vertices, with the remaining part 'suspended.' That alone does not force the non-bipartite part to be a single K_r attached to the smaller side of the bipartite Turán graph. The eigenvalue comparison has to use exact attachment, not just vertex count. The stress-test note makes this point correctly. I suspect the full paper does contain such a lemma, but it is not visible here.\n\nThe citation pattern looks honest: prior work by Ren-Wang-Wang-Yang, Yan-Peng, Guo-Lin-Zhao, Zhang-Zhao is correctly positioned. No fitted constants or circularity are apparent. The condition n ≥ 712k is large but linear, in line with the claimed improvement.\n\nBottom line: this deserves a serious referee. The claims are important enough that the referee time is justified even if the final verdict is uncertain. If the exact-stability lemma holds, it is a strong paper. I would not desk reject.","headline":"A credible abstract claiming a real improvement in spectral extremal graph theory, but the load-bearing exact-stability lemma is hidden in the full text.","tokens_in":1969,"tokens_out":1483,"would_cite":false,"duration_ms":14973,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C35","05C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a sharp spectral-radius bound for C_{2k+1}-free graphs with chromatic number at least r, with equality only for the graph formed by joining a K_r to the smaller part of a complete bipartite graph.","keywords":["spectral radius","C_{2k+1}-free graphs","chromatic number","extremal graph theory","stability","bipartite Turán graph","spectral extremal problem","odd cycles"],"falsifier":"A concrete way to test the claim is to search, by computation, for an n-vertex C_{2k+1}-free graph with χ(G) ≥ r and n ≥ 712k whose adjacency spectral radius exceeds λ(T_{n-r+1,2}∘K_r); for instance, attaching the clique K_r to the larger partite set rather than the smaller one, or adding a single extra edge inside the bipartite core, would yield a candidate counterexample if its spectral radius is larger.","tokens_in":1035,"feed_emoji":"📐","tokens_out":4288,"duration_ms":37161,"temperature":0.7,"pith_summary":"This paper proves a sharp spectral-radius bound for graphs that contain no odd cycle of length 2k+1 yet require many colors: if such a graph has chromatic number at least r, its spectral radius cannot exceed that of the graph obtained by gluing a clique K_r onto the smaller side of a nearly balanced complete bipartite graph. Equality holds only for that construction. The proof rests on a new structural stability theorem, which says that once n is at least linear in k, every sufficiently dense odd-cycle-free graph of chromatic number at least r consists of a large bipartite core with at most r-2 additional vertices suspended over it. This gives the first solution to the spectral extremal problem for F-free graphs with high chromatic number.","feed_headline":"Sharp spectral bound for high-chromatic odd-cycle-free graphs","feed_subtitle":"Extremal graph is a complete bipartite graph with a clique attached to one side; equality is unique.","key_machinery":"The load-bearing tool is a structural stability decomposition: every n-vertex C_{2k+1}-free graph with chromatic number at least r, for n at least linear in k, can be obtained from a large bipartite graph by suspending small graphs whose total vertex count is at most r−2. This is what converts the extremal problem into an eigenvalue comparison on a bounded-modification bipartite graph, and it is also what forces the extremal configuration to be the clique attachment T_{n-r+1,2}∘K_r.","core_discovery":"The central claim is the spectral theorem: for 3 ≤ r ≤ 2k and n ≥ 712k, if G is an n-vertex C_{2k+1}-free graph with χ(G) ≥ r, then λ(G) ≤ λ(T_{n-r+1,2}∘K_r), with equality if and only if G = T_{n-r+1,2}∘K_r. The extremal graph is a complete bipartite Turán graph with an additional clique identifying a vertex of the clique with a vertex of the smaller bipartition class. This resolves the spectral extremal question for F-free graphs of high chromatic number in this family, extending earlier results that covered special cases.","pith_inferences":["The linear dependence on k suggests that a similar decomposition might hold for other families of graphs defined by forbidden odd cycles or color-critical graphs, but this is not claimed by the paper.","The choice of identifying the clique with the smaller partite set may be deliberate because it balances the spectral contributions; swapping to the larger side may produce a graph with smaller or larger radius depending on the parameters, a question not explored here.","The threshold 712k is likely not optimal; one could test numerically whether smaller constants work for specific k and r, but the paper does not pursue that."],"forward_implications":["If the spectral theorem holds, then for every r ≤ 2k the spectral radius of C_{2k+1}-free graphs of chromatic number at least r is maximized by the same explicit construction.","The structural result yields a tight upper bound on the number of edges in such graphs, recovering and improving the previous quadratic bound with a linear condition on n.","The equality case pins down the extremal graph uniquely, showing that any other graph with the same property is strictly spectrally smaller.","The work provides the first solution to the spectral extremal problem for F-free graphs with high chromatic number in this setting.","The linear bound n ≥ 712k is a quantitative improvement over earlier constraints that required n to grow quadratically in r and k."],"supporting_citations":[],"fun_headline_variants":["Sharp spectral bound for odd-cycle-free high-chromatic graphs","Unique spectral extremal: bipartite graph with a clique attached","Tight spectral radius for high-chromatic graphs without odd cycles","Spectral extremal solved for odd-cycle-free high-chromatic graphs","Odd-cycle-free graphs: sharp spectral bound with unique extremal"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The entire proof depends on the structural stability theorem from the first part: that every C_{2k+1}-free graph with chromatic number at least r has a large bipartite core with at most r−2 suspended vertices once n is at least linear in k; if this decomposition admits exceptions, the spectral comparison and the equality case collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sharp spectral bound for odd-cycle-free high-chromatic graphs","Unique spectral extremal: bipartite graph with a clique attached","Tight spectral radius for high-chromatic graphs without odd cycles","Spectral extremal solved for odd-cycle-free high-chromatic graphs","Odd-cycle-free graphs: sharp spectral bound with unique extremal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000806,"raw_usage":{"total_tokens":3531,"prompt_tokens":1052,"completion_tokens":2479,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":796,"completion_tokens_details":{"reasoning_tokens":2393}},"tokens_in":796,"tokens_out":2479,"duration_ms":17781,"temperature":1.0,"reasoning_tokens":2393,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:57:22.972698+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the claim is to search, by computation, for an n-vertex C_{2k+1}-free graph with χ(G) ≥ r and n ≥ 712k whose adjacency spectral radius exceeds λ(T_{n-r+1,2}∘K_r); for instance, attaching the clique K_r to the larger partite set rather than the smaller one, or adding a single extra edge inside the bipartite core, would yield a candidate counterexample if its spectral radius is larger.","supporting_citations":[],"review_version":1}