{"id":"c9448cb0-e2a3-4259-92c6-52a72a637fec","arxiv_id":"2509.00769","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For b > a, every connected n-vertex graph with minimum degree at least a and spectral radius at least rho(H^{a,b}_n) contains an [a,b]-factor, except H^{a,b}_n itself.","lead":"This paper proves a sharp spectral threshold: any connected graph with large minimum degree and a sufficiently large largest eigenvalue must contain a spanning subgraph where every vertex has between a and b neighbors, unless it is exactly one exceptional graph. It resolves an open problem posed by Hao and Li and extends a previous result that handled the special case a equals b.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the suspected Lemma 2.4 issue and Case 2.2 inequality direction are not actual flaws.","rationale":"The reader's weakest assumption targets Lemma 2.4 and the Case 2.2 inequality direction. I checked both carefully. Lemma 2.4 matches the known Heinrich et al. criterion for g<f, so the reliance is legitimate. The Case 2.2 display, when read as ∑d ≤ a|W|−b|S|−1, is precisely the negation of the lemma's sufficient condition and is correctly applied. The real issues are localized typos: the variable typo in Lemma 2.7 and the slightly too strong ρ(G)>n−b−1 statement in Case 2. Neither is load-bearing, since ρ(G)>n−b−2 suffices for the needed strict positivity. The long spectral comparison in Lemma 2.7 has several dense algebraic steps, but the expansion and lower-bounding of f(ρ,ρ′) check out under the stated inequalities q≤a−1, p1≤a−2, and n≥2(a+b+2)(b+2). Overall, the central claim is supported by a coherent extremal argument, and the reader's conditional verdict can remain unchanged; the paper only needs minor typographical corrections, not substantive revision.","tokens_in":11214,"tokens_out":31680,"duration_ms":335461,"concrete_test":"Independently verify Lemma 2.4 by brute-forcing all graphs up to n=8 for a few pairs with b>a (e.g., a=1,b=2 and a=1,b=3), comparing the single-set inequality against exhaustive [a,b]-factor existence; this directly settles the only foundational criterion the proof depends on and would confirm the direction used in Case 2.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central argument of Theorem 1.1. Lemma 2.4 is the standard Heinrich–Hell–Kirkpatrick–Liu single-set criterion for (g,f)-factors with g<f; for [a,b] with a<b it is an exact characterization, so the proof's reliance on it is sound. The 'suspicious inequality direction' in Subcase 2.2 is not an error: the display 0=∑_{v∈W2} d_{G*−S2}(v) ≤ a|W2|−b|S2|−1 = a−1 is exactly the violation form of Lemma 2.4, a|W|−∑d ≥ b|S|+1, and therefore correctly implies that G* has no [a,b]-factor. The only genuine defects I found are a variable typo in Lemma 2.7 ('p(p1+a+1)' should be 'q(p1+a+1)') and a harmless overstatement in Case 2 where ρ(G)>n−b−1 would better read n−b−2; neither affects the argument because ρ(G)>n−b−2>a−1≥q is sufficient. The spectral comparisons, case analysis, and final contradiction in Subcase 2.2 are consistent after restoring the missing q and using the correct lower bound. No internal inconsistency or circular step surfaced.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a sharp spectral-radius condition for the existence of an [a,b]-factor in connected graphs with minimum degree at least a, for b > a >= 1. The main result, Theorem 1.1, states that for n >= 2(a+b+2)(b+2), if rho(G) >= rho(H^{a,b}_n), then G contains an [a,b]-factor unless G is isomorphic to H^{a,b}_n, a specific graph shown to contain no such factor. The proof combines Perron-vector edge-switching arguments, the Hong-Shu-Fang/Nikiforov spectral-radius bound, the Heinrich et al. single-set version of Tutte's factor theorem, and a detailed case analysis on the vertex set W appearing in the factor-criterion violation. The paper also states a size version and a fractional-factor corollary.","tokens_in":11546,"tokens_out":18429,"duration_ms":191018,"significance":"If the proof is correct, the paper resolves the spectral-radius part of Problem 1 posed by Hao and Li, complementing the known a = b case. The extremal graph is natural and the condition is shown to be sharp. A particular strength is that the extremal graph is not assumed but constructed from the factor-criterion violation, and the proof relies on established external lemmas rather than circular reasoning. The main argument is technically demanding; I could verify the overall structure and the key reduction steps, and the potential issues flagged in the reading report (Lemma 2.4 applicability and the final Subcase 2.2 inequality) are not actual mathematical flaws. However, several typos and terse inequality checks make the paper harder to read and should be corrected.","major_comments":[],"minor_comments":[{"comment":"In the displayed definition of f(rho,rho'), the term '(rho^2 + rho + p(p1 + a + 1))' should read 'q(p1 + a + 1)'. The preceding denominator and inequality (5) use q, and the later bounds rely on 2 <= q <= a-1. As printed, the expression is inconsistent and cannot be verified. Please correct this typo throughout the displayed formula.","section":"Lemma 2.7 (p. 5)"},{"comment":"The sentence 'we have rho(G) > n - b - 1 > a - 1 >= q' overstates what is known: (13) gives only rho(G) > n - b - 2. The needed conclusion is rho(G) > n - b - 2 > a - 1 >= q, which is sufficient. Please correct the displayed inequality.","section":"Theorem 1.1, Case 2 (p. 7)"},{"comment":"The final check that G* has no [a,b]-factor is written as '0 = sum_{v in W2} d_{G*-S2}(v) <= a|W2| - b|S2| - 1 = a - 1'. This is confusing because Lemma 2.4 requires the opposite inequality, a|W2| - sum d_{G*-S2}(v) >= b|S2| + 1, to certify non-existence of a factor. Since d_{G*-S2}(v)=0 for every v in W2, the violation condition is a(b+1) >= ab+1, i.e. a >= 1; the displayed line is an equivalent rearrangement but the implication should be stated explicitly. Please rewrite this step.","section":"Theorem 1.1, Subcase 2.2 (p. 8-9)"},{"comment":"The deduction 'by the maximality of rho(G), G[V(G)\\ W] is K_{n-t} and e(S,W)=st' is used later in the proof but is not justified. Please add a sentence explaining that adding any missing edge with at least one endpoint outside W (or between S and W) does not alter the certificate from Lemma 2.4 for S and W, so the no-factor property and the minimum-degree condition are preserved, contradicting maximality. This is a small but important justification.","section":"Theorem 1.1, after (13)"},{"comment":"The statement 'H^{a,b}_n contains no [a,b]-factor' is used to justify rho(G) >= rho(H^{a,b}_n) and is plausible, but no proof is given in the text. A one-line counting argument (the b+1 vertices in W have total degree demand a(b+1), while the capacity from the a vertices of K_a plus the a-1 extra edges is at most ab+a-1) would make the sharpness claim self-contained.","section":"Theorem 1.1, after (13)"}],"recommendation":"minor_revision","confidential_remarks":"The paper appears technically sound after correcting the identified typos and clarifying the final factor-criterion check. The proof is long and somewhat calculation-heavy; the authors should carefully proofread the displayed inequalities in Lemma 2.7 and in Case 2/Subcase 2.2. The contribution is appropriate for a note and resolves a stated open problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline is that this is a solid, incremental resolution of a named open problem. Fan, Lin and Zhu prove a sharp threshold: for connected n-vertex graphs with δ≥a and n large, ρ(G)≥ρ(H^{a,b}_n) forces an [a,b]-factor unless G is H^{a,b}_n. That graph, built from K_a joined to a large clique plus b+1 isolated vertices, is genuinely new and the bound is tight. This covers exactly the b>a case that Tang-Zhang's a=b result leaves open, so the problem from Hao-Li is answered.\n\nThe paper does what it claims. It uses the standard single-set Heinrich et al. factor criterion, Perron-vector switching, and the Hong-Shu-Fang/Nikiforov bound. The structure is clear, and the citations to Hao-Li and Tang-Zhang are appropriate. I did not verify every inequality, but I checked the logic at the joints and found no load-bearing flaw.\n\nThe soft spots are minor and fixable. Lemma 2.7 has a typo: 'p(p1+a+1)' should be 'q(p1+a+1)' in the displayed definition of f(ρ,ρ'). In Case 2, the write-up says ρ(G)>n−b−1 where n−b−2 would be the correct strict lower bound; this is harmless because n−b−2>a−1. The reader's worry about the inequality direction in Subcase 2.2 is, on reading, not real: 0=∑_{v∈W2} d_{G*−S2}(v) ≤ a|W2|−b|S2|−1 = a−1 is exactly the violation of Lemma 2.4 for a≥1, so it correctly shows G* has no [a,b]-factor.\n\nThe bigger soft spot is Theorem 3.1 in the concluding remarks. It states a size threshold without proof, citing only 'a technique analogous to [17]'. That is not enough. Either give the proof or downgrade it to a remark/conjecture.\n\nWho is this for? Anyone working on spectral conditions for factors. It is not a conceptual breakthrough, but it is a correct-looking resolution of an open problem with the right extremal construction. A serious referee should go through the algebra in Lemma 2.7 carefully; the paper deserves that effort. I would send it out, and require the typo fix and the Theorem 3.1 issue addressed before publication.","headline":"Solves the b>a spectral radius case of Hao-Li's factor problem with a sharp extremal graph; the proof is standard but dense, with minor typos and one unproved side theorem.","tokens_in":12018,"tokens_out":4069,"would_cite":true,"duration_ms":45948,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"A sharp spectral radius threshold guarantees [a,b]-factors in large connected graphs with minimum degree at least a.","keywords":["spectral radius","[a,b]-factor","extremal graph","minimum degree","fractional [a,b]-factor","graph eigenvalues","sharp bound"],"falsifier":"Constructing a connected graph G with n≥2(a+b+2)(b+2), δ(G)≥a, ρ(G)≥ρ(H^{a,b}_n), and no [a,b]-factor, with G not isomorphic to H^{a,b}_n, would disprove Theorem 1.1. A more targeted check is to verify Lemma 2.4 against a graph that satisfies the displayed inequality but is known to lack an [a,b]-factor for b>a; if such a graph exists, the proof's central criterion fails.","tokens_in":11138,"feed_emoji":"🧩","tokens_out":4506,"duration_ms":48398,"temperature":0.7,"pith_summary":"This paper establishes a sharp spectral-radius condition for the existence of an [a,b]-factor: for b>a≥1, every connected n-vertex graph with minimum degree at least a and n at least 2(a+b+2)(b+2) has an [a,b]-factor as soon as its spectral radius reaches that of a specific extremal graph H^{a,b}_n. The only exception is H^{a,b}_n itself, which is shown to contain no [a,b]-factor. The result answers an open problem about sharp bounds for [a,b]-factors under a minimum-degree assumption. Because a graph with minimum degree below a cannot contain such a factor, the degree hypothesis is natural, and the extremal graph makes the threshold tight.","feed_headline":"Spectral radius threshold forces [a,b]-factors in large graphs","feed_subtitle":"In graphs with minimum degree at least a, beating the extremal graph H forces a spanning factor unless the graph is H itself.","key_machinery":"The argument runs on three tools. First, a single-set [a,b]-factor criterion (Lemma 2.4) that characterizes factor existence through an inequality involving a set S, the vertices of low degree in G−S, and a,b. Second, a spectral perturbation rule (Lemma 2.1) that compares spectral radii after moving edges from a lower Perron-coordinate vertex to a higher one; this lets the proof deform any maximal factor-free graph into a canonical shape. Third, Lemmas 2.6 and 2.7 identify H^{a,b}_n as the unique spectral-radius maximizer within the relevant family of factor-free candidates. The order lower bound n≥2(a+b+2)(b+2) is used throughout to control the upper-bound estimates.","core_discovery":"The central claim is Theorem 1.1. Let a and b be positive integers with b>a, and let G be a connected graph of order n≥2(a+b+2)(b+2) with minimum degree δ(G)≥a. If ρ(G)≥ρ(H^{a,b}_n), then G contains an [a,b]-factor unless G is isomorphic to H^{a,b}_n. The graph H^{a,b}_n is built by taking a join K_a ∨ (K_{n-a-b-1} ∪ (b+1)K_1) and adding a−1 edges from one isolated vertex to a−1 vertices in the large clique. H^{a,b}_n has no [a,b]-factor, so the spectral threshold cannot be lowered. The proof maximizes spectral radius among factor-free graphs, uses a single-subset version of the [a,b]-factor existence criterion, and shows that any graph beating the threshold must satisfy that criterion.","pith_inferences":["A natural next step is to test whether the order threshold 2(a+b+2)(b+2) can be substantially lowered; the proof's estimates likely leave room for a smaller true extremal order.","The same extremal construction may guide sharp spectral conditions for related factor types, such as [a,b]-parity factors, where the factor criterion changes.","One could probe whether replacing the minimum-degree condition δ(G)≥a with a weaker average-degree hypothesis changes the extremal graph or the threshold."],"forward_implications":["Every connected graph with minimum degree at least a and order above the stated threshold either contains an [a,b]-factor or is exactly the extremal graph H^{a,b}_n.","The threshold ρ(H^{a,b}_n) is sharp: H^{a,b}_n attains it but has no [a,b]-factor.","The same spectral condition also forces a fractional [a,b]-factor, since every integral [a,b]-factor is fractional.","A companion size condition in the paper gives a sharp edge-count threshold for the same factor problem.","This resolves the spectral-radius side of the open problem by supplying the sharp lower bound."],"supporting_citations":[{"why":"Supplies the single-set [a,b]-factor existence criterion used as Lemma 2.4, the foundation of the whole extremal argument.","marker":"[11]"},{"why":"Provides the spectral perturbation lemmas (Lemma 2.1 and 2.5) used to deform a maximal factor-free graph toward the extremal configuration.","marker":"[21]"},{"why":"Gives the sharp upper bound on spectral radius used to estimate ρ(G) from edge count and minimum degree.","marker":"[14]"},{"why":"Independently gives the same spectral-radius upper bound and the monotonicity property of the bounding function used in the estimates.","marker":"[22]"},{"why":"Poses the open problem that this paper resolves for the spectral-radius side.","marker":"[12]"},{"why":"Treats the a=b case of the same problem, the result this paper extends to b>a.","marker":"[26]"}],"fun_headline_variants":["Spectral radius threshold forces [a,b]-factors","Eigenvalue bound guarantees [a,b]-factor existence","New spectral extremal result for [a,b]-factors","Resolving Hao-Li problem on spectral radius","Spectral radius criterion for [a,b]-factors"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof relies on the single-set [a,b]-factor criterion being exactly right for all b>a; if that criterion admits exceptions, the step where a factor-free graph must violate the inequality falls apart.","fun_headline_variants_meta":{"raw":{"variants":["Spectral radius threshold forces [a,b]-factors","Eigenvalue bound guarantees [a,b]-factor existence","New spectral extremal result for [a,b]-factors","Resolving Hao-Li problem on spectral radius","Spectral radius criterion for [a,b]-factors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1059,"prompt_tokens":700,"completion_tokens":359,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":444,"tokens_out":359,"duration_ms":4445,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:13:21.338832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Constructing a connected graph G with n≥2(a+b+2)(b+2), δ(G)≥a, ρ(G)≥ρ(H^{a,b}_n), and no [a,b]-factor, with G not isomorphic to H^{a,b}_n, would disprove Theorem 1.1. A more targeted check is to verify Lemma 2.4 against a graph that satisfies the displayed inequality but is known to lack an [a,b]-factor for b>a; if such a graph exists, the proof's central criterion fails.","supporting_citations":[{"cited_title":"Heinrich, P","cited_arxiv_id":null,"evidence_quote":"Supplies the single-set [a,b]-factor existence criterion used as Lemma 2.4, the foundation of the whole extremal argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the sharp upper bound on spectral radius used to estimate ρ(G) from edge count and minimum degree."},{"cited_title":"Nikiforov, Some inequalities for the largest eigenvalue of a graph, Combin","cited_arxiv_id":null,"evidence_quote":"Independently gives the same spectral-radius upper bound and the monotonicity property of the bounding function used in the estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Poses the open problem that this paper resolves for the spectral-radius side."},{"cited_title":"Spectral conditions for graphs to contain $k$-factors","cited_arxiv_id":"2508.05678","evidence_quote":"Treats the a=b case of the same problem, the result this paper extends to b>a."}],"review_version":1}