{"id":"d43714d8-0652-49ba-b3bf-e23fde5ee69a","arxiv_id":"2508.05678","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The claimed sharp spectral condition for k-factors fails because the extremal graph G_{n,k} itself has a k-factor.","lead":"This paper claims a sharp spectral radius threshold for a graph to contain a k-factor, for every k at least 2. The proposed extremal graph G_{n,k}, used to show sharpness, actually contains such a k-factor, so the proof and the stated exception collapse.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed extremal graph G_{n,k} actually has a k-factor, so the sharpness example and the uniqueness lemmas fail; Theorem 1.1 is unsupported.","rationale":"The reader's criticism is correct and load-bearing. The paper's main theorem is advertised as a sharp spectral condition, and sharpness rests entirely on the assertion that G_{n,k} has no k-factor. Since G_{n,k} actually has a k-factor, the claimed extremal counterexample does not exist. Worse, the false assertion is used in the proof infrastructure: Lemma 3.1 and Lemma 4.1 both require G_{n,k} to lie in the relevant obstruction sets, but the degree-sum computations with the stated S and T fail by a margin of roughly k^2+k. These are not typographical slips that can be repaired locally; the structural Claim 2 in Lemma 3.1 forces a degree sum in B that contradicts the defining inequality of G_k^n. I therefore agree with the reader's REJECT verdict and see no reason to change it. I am not claiming the general spectral-factor approach is worthless, only that the manuscript as written is not correct.","tokens_in":10570,"tokens_out":12737,"duration_ms":151706,"concrete_test":"Construct the k-factor explicitly: take all edges of the K_{k+1} clique, and on the remaining n−k−1 vertices — which form K_{n−k−1} — take any k-regular spanning subgraph, which exists because k(n−k−1) is even and n−k−1≥k+1. Verify that every vertex of G_{n,k} has degree exactly k in the union. This single construction disproves the paper's assertion that G_{n,k} is k-factor-free. A second, independent check is to recompute the two displayed membership inequalities; both fail for k≥2, e.g. k=2, n=10 gives left side 13 versus right side 1 in the Lemma 4.1 check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 1 defines G_{n,k}=K_k∨(K_{k+1}∪K_{n−1−2k}) plus k−1 edges from one vertex of K_{k+1} to K_{n−1−2k}, and asserts that it contains no k-factor. This assertion is false. The clique K_{k+1} is k-regular by itself. The remaining vertices W∪V induce the complete graph K_{n−k−1}, because K_k and K_{n−1−2k} are complete and the join supplies all W–V edges. Since kn is even, k(n−k−1)=kn−k(k+1) is even, and n−k−1≥k+1 under the theorem's hypotheses; hence K_{n−k−1} has a k-factor. The union of this factor with E(K_{k+1}) is a k-factor of G_{n,k}. Thus the 'unless G=G_{n,k}' exception is not a genuine exception, and the claimed sharp bound has no extremal counterexample. The same error propagates into the membership checks: in Lemma 3.1, with B=V(K_{k+1}), ∑_{u∈B} d_{G_{n,k}}(u)=k(k+1)+k(k+1)+(k−1)=2k^2+3k−1, not ≤k^2+2k−1; in Lemma 4.1, with S=V(K_k), T=V(K_{k+1}), the left side is k^2+2k−1 while k|T|−k|S|−2+q=k−1. So the assertions 'G_{n,k}∈G_k^n' and 'G_{n,k}∈G_n,k' fail as written. Lemma 3.1 and Lemma 4.1 therefore cannot support Theorem 1.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates spectral radius conditions that force the existence of a k-factor in an n-vertex graph with minimum degree at least k. For k ≥ 2, kn even, and n ≥ max{k^2+6k+7, 20k+10}, the authors define an extremal graph G_{n,k} = K_k ∨ (K_{k+1} ∪ K_{n−1−2k}) with k−1 extra edges from one vertex of K_{k+1} to vertices of K_{n−1−2k}, assert that this graph has no k-factor, and prove Theorem 1.1: if ρ(G) ≥ ρ(G_{n,k}), then G has a k-factor unless G = G_{n,k}. The proof uses Tutte's f-factor theorem to translate the absence of a k-factor into membership in a class of graphs, then invokes spectral perturbation lemmas to identify the spectral extremal graph in that class.","tokens_in":10887,"tokens_out":6241,"duration_ms":67876,"significance":"A correct theorem of this type would be a meaningful contribution and would resolve Problem 1 of Fan and Lin for all k ≥ 2. The paper uses standard and appropriate tools — Tutte's f-factor theorem, the Hong–Shu–Fang spectral bound, Perron–Frobenius theory, and an edge-swap lemma — in a self-contained way, which is a strength. However, the central extremal construction is invalid: the asserted factor-free graph G_{n,k} in fact contains a k-factor. As a result, the claimed sharpness of the spectral bound and the uniqueness lemmas that support Theorem 1.1 are not established. The manuscript as it stands does not deliver its central claim.","major_comments":[{"comment":"The assertion that G_{n,k} contains no k-factor is false. The clique on V(K_{k+1}) is k-regular by itself, and the remaining n−k−1 vertices induce the complete graph K_{n−k−1}: the sets V(K_k) and V(K_{n−1−2k}) are internally complete and are joined to each other. Since k(n−k−1) = kn − k(k+1) is even (because kn is even and k(k+1) is even) and n−k−1 ≥ k+1 under the theorem's hypotheses, K_{n−k−1} has a k-factor. The union of that factor with E(K_{k+1}) is a k-factor of G_{n,k}. The counting argument in the introduction is incorrect because it ignores that each vertex of K_{k+1} already achieves degree k within that clique. Therefore G_{n,k} cannot serve as the sharpness example, and the exception in Theorem 1.1 is not a genuine exception.","section":"Section 1, definition of G_{n,k}"},{"comment":"The claim 'Clearly, G_{n,k} ∈ G_k^n' is false. For B = V(K_{k+1}), each vertex in B has degree 2k inside G_{n,k} (k neighbors in K_{k+1} and k neighbors in K_k), except the one distinguished vertex that has k−1 additional edges to K_{n−1−2k}. Hence ∑_{u∈B} d_{G_{n,k}}(u) = (k+1)·2k + (k−1) = 2k^2 + 3k − 1, which exceeds the threshold k^2 + 2k − 1 in the definition of G_k^n. Consequently Lemma 3.1 does not apply to G_{n,k}, and its conclusion cannot identify G_{n,k} as the extremal graph in G_k^n.","section":"Section 3, Lemma 3.1"},{"comment":"The assertion 'It is easy to check that G_{n,k} ∈ G_n,k' is false. Taking S = V(K_k) and T = V(K_{k+1}), the left-hand side of the defining inequality is ∑_{u∈T} d_{G−S}(u) = k(k+1) + (k−1) = k^2 + 2k − 1, while the right-hand side is k|T| − k|S| − 2 + q = k(k+1) − k^2 − 2 + 1 = k − 1. For k ≥ 2, the inequality fails. Thus G_{n,k} is not a member of the class G_n,k, and the final step of Theorem 1.1's proof, which concludes G = G_{n,k} from Lemma 4.1, does not go through. The proof of Theorem 1.1 is therefore unsupported.","section":"Section 4, Lemma 4.1"}],"minor_comments":[{"comment":"The title reads 'Spectral conditions for graphs to containk-factors'; there should be a space before 'k-factors'.","section":"Title"},{"comment":"In the displayed chain after inequality (3), the expression 't(n−t)t' appears to be a typo; it should presumably be 't(n−t)'.","section":"Section 4, Subcase 1.2"},{"comment":"The notation is inconsistent: the extremal graph is usually G_{n,k}, but the class of graphs in Section 4 is also denoted G_n,k. This makes statements such as 'G_{n,k} ∈ G_n,k' harder to read and should be clarified.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The error in the definition of G_{n,k} is easily verified mathematically and is load-bearing: both extremal lemmas use the false membership of G_{n,k} in their respective classes, and the sharpness claim depends on the false assertion that G_{n,k} has no k-factor. This is not a presentation issue; repairing it would require a different extremal construction and a reproof of the main theorem. I recommend rejection, though a substantially revised manuscript with a valid extremal graph could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: the main theorem is unsupported because the extremal graph G_{n,k} actually has a k-factor. The graph is K_k ∨ (K_{k+1} ∪ K_{n−1−2k}) plus k−1 edges from one vertex of K_{k+1}. The clique K_{k+1} is already k-regular, and the remaining k+(n−1−2k)=n−1−k vertices induce a complete graph K_{n−1−k}. That complete graph has a k-factor whenever k(n−1−k) is even; here kn even implies k(n−1−k)=kn−k(k+1) even, and n−1−k ≥ k+1 holds under the stated n bound. So a k-factor exists. The introduction's counting argument ignores that factor edges can stay inside K_{k+1}. This invalidates the claimed sharpness example, the membership checks in Lemmas 3.1 and 4.1, and the uniqueness conclusions that lean on them.\n\nWhat is genuinely new: the problem is a natural spectral version of Fan–Lin's binding-number question for all k ≥ 2, and the proof architecture—use Tutte's f-factor theorem to force membership in a constrained graph class, then identify the spectrally extremal member—is reasonable. The edge-swapping argument in Lemma 3.1 is standard and may be repairable. Citations to the relevant spectral factor literature are appropriate. Conditional on a correct extremal graph, the approach could work.\n\nSoft spots beyond the fatal one: equation (1) claims e(G) < (k+1)n − (k+1)^2, but G_{n,k} itself has roughly n^2/2 edges, so that inequality is false as written (likely e(\\bar{G}) was meant). There is also an 's ≥ k' typo in Lemma 3.1. These are secondary next to the failure of the extremal construction.\n\nBottom line: this is a serious attempt at a legitimate problem, but the central example fails, so the theorem as stated is false. I would not cite it. If an editor wants a referee to document the counterexample, that is reasonable, but I would not expect the paper to survive in anything like its current form. A corrected version with a different no-k-factor extremal graph might deserve another look.","headline":"The extremal graph G_{n,k} actually has a k-factor, so the sharpness example and the main theorem collapse; the proof structure is plausible but the central construction is wrong.","tokens_in":11470,"tokens_out":6697,"would_cite":false,"duration_ms":66408,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"A sharp spectral-radius threshold guarantees k-factors in all sufficiently large graphs of minimum degree at least k.","keywords":["spectral radius","k-factor","adjacency matrix","minimum degree","extremal graph","binding number","join of cliques","f-factor theorem"],"falsifier":"Construct an explicit k-factor of G_{n,k} for any admissible k and n, say k = 2 and n = 50: the clique K_{k+1} alone is k-regular and the remaining clique has order at least k + 1, so a spanning k-regular subgraph can be assembled by combining a k-factor of K_{k+1} with a k-factor of the large clique. If such a construction succeeds, the sharpness assertion of the paper is false even though the sufficiency theorem may remain true.","tokens_in":10300,"feed_emoji":"📐","tokens_out":10653,"duration_ms":119399,"temperature":0.7,"pith_summary":"This paper tries to identify, by its spectral radius alone, when a graph is guaranteed to contain a k-factor, a spanning subgraph in which every vertex has degree exactly k. The proposed answer is a sharp threshold: for k ≥ 2, kn even, and n at least roughly k² + 6k + 7 or 20k + 10, every n-vertex graph with minimum degree at least k and spectral radius at least ρ(G_{n,k}) must contain a k-factor, unless it is exactly the extremal graph G_{n,k}. The extremal graph is built from three cliques: K_k joined to K_{k+1} ∪ K_{n−1−2k}, with k−1 extra edges from one vertex of the middle clique to vertices of the largest clique. If the theorem is right, checking one eigenvalue settles a problem on 1-binding graphs that was previously open for general k.","feed_headline":"Spectral radius above one graph's value forces a k-factor","feed_subtitle":"For every k≥2 and large n, one extremal graph sets the eigenvalue threshold for a k-regular spanning subgraph.","key_machinery":"The machinery has three parts. The first is the f-factor theorem in the form of Lemma 2.4, which converts 'no k-factor' into an inequality on pairs of disjoint vertex sets S and T. The second is a spectral edge-switching lemma that compares Perron-vector entries and says that moving an edge from a smaller-weight position to a larger-weight position strictly increases the spectral radius; this is what forces the extremal obstruction to be a join of cliques with neatly ordered Perron entries. The third is the extremal object itself, G_{n,k}, the join K_k ∨ (K_{k+1} ∪ K_{n−1−2k}) with k−1 added edges; its spectral radius is the threshold that runs through the whole statement.","core_discovery":"The central discovery is that the spectral radius of the specific graph G_{n,k} = K_k ∨ (K_{k+1} ∪ K_{n−1−2k}) plus k−1 cross edges is the cutoff for k-factor existence among n-vertex graphs with minimum degree at least k. The proof works by contraposition: if G has no k-factor, the f-factor theorem supplies disjoint vertex sets S, T whose counts violate the factor condition; that violation puts G into a family of obstruction graphs. An edge-switching argument with Perron vectors shows that within this family the spectral radius is maximized uniquely by G_{n,k}. Hence any graph with ρ(G) ≥ ρ(G_{n,k}) cannot be a non-factor graph except possibly G_{n,k} itself, which the paper asserts has no k-factor; this last assertion is what makes the bound sharp.","pith_inferences":["The theorem's proof does not actually seem to need the extremal graph G_{n,k} to be factor-free; if G_{n,k} turns out to contain a k-factor, the sufficiency result would survive but the word 'sharp' would have to be withdrawn.","A concrete check for k = 2, n = 50 would settle the sharpness question immediately: K_3 is already a 2-factor on its own and the remaining large clique has a 2-factor, so an explicit 2-factor of G_{n,2} can likely be written down; that would contradict the paper's claimed sharpness.","The same obstruction-family-plus-spectral-maximization scheme could be adapted to [a,b]-factors or to odd factors, where the parity term in the f-factor theorem changes and may produce different extremal graphs."],"forward_implications":["For every k ≥ 2, a single number ρ(G_{n,k}) decides k-factor existence for all large graphs with minimum degree at least k: if the spectral radius is at least that number, a k-factor is guaranteed.","The theorem solves the non-bipartite case of the binding-number problem for every k ≥ 2, since G_{n,k} is 1-binding.","The structural conclusion is rigid: any graph that reaches the threshold and still lacks a k-factor would have to be isomorphic to G_{n,k}.","The threshold is claimed to be best possible, because G_{n,k} itself is asserted to contain no k-factor; if that assertion holds, no smaller constant can replace ρ(G_{n,k}).","The evenness condition kn ≡ 0 mod 2 is built into the statement, so the result also covers every k ≥ 2 for which a k-factor can exist at all."],"supporting_citations":[{"why":"Poses the binding-number problem that this paper answers for all k ≥ 2 and supplies the k = 1, 2 spectral characterization that the result extends.","marker":"[4]"},{"why":"Source of the spectral monotonicity lemma (Lemma 2.1) used to show the extremal graph is edge-maximal.","marker":"[1]"},{"why":"Provides the sharp upper bound on spectral radius in terms of degree sequence and edge count (Lemma 2.3) used to bound non-extremal candidates.","marker":"[7]"},{"why":"The f-factor theorem (Lemma 2.4) is the exact criterion that turns absence of a k-factor into the obstruction-family inequality.","marker":"[11]"},{"why":"Supplies the edge-switching lemma (Lemma 2.2) that compares spectral radii of graphs differing by a few edges, the main tool in the extremal maximization.","marker":"[13]"}],"fun_headline_variants":["Sharp spectral radius cutoff for k-factors revealed","One extremal graph sets eigenvalue threshold for k-factors","Spectral bound guarantees k-regular spanning subgraph","k-factor existence pinned by spectral radius threshold","Eigenvalue condition forces k-factors in large graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharpness of the bound rests on the claim that the extremal graph G_{n,k} has no k-factor; the paper's proof of that claim counts only edges leaving the middle clique and misses that K_{k+1} itself is already k-regular.","fun_headline_variants_meta":{"raw":{"variants":["Sharp spectral radius cutoff for k-factors revealed","One extremal graph sets eigenvalue threshold for k-factors","Spectral bound guarantees k-regular spanning subgraph","k-factor existence pinned by spectral radius threshold","Eigenvalue condition forces k-factors in large graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2820,"prompt_tokens":851,"completion_tokens":1969,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":1894}},"tokens_in":467,"tokens_out":1969,"duration_ms":16264,"temperature":1.0,"reasoning_tokens":1894,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:34:18.344974+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an explicit k-factor of G_{n,k} for any admissible k and n, say k = 2 and n = 50: the clique K_{k+1} alone is k-regular and the remaining clique has order at least k + 1, so a spanning k-regular subgraph can be assembled by combining a k-factor of K_{k+1} with a k-factor of the large clique. If such a construction succeeds, the sharpness assertion of the paper is false even though the sufficiency theorem may remain true.","supporting_citations":[{"cited_title":"Fan and H","cited_arxiv_id":null,"evidence_quote":"Poses the binding-number problem that this paper answers for all k ≥ 2 and supplies the k = 1, 2 spectral characterization that the result extends."},{"cited_title":"Hong, J.L","cited_arxiv_id":null,"evidence_quote":"Provides the sharp upper bound on spectral radius in terms of degree sequence and edge count (Lemma 2.3) used to bound non-extremal candidates."},{"cited_title":"Tutte, The factors of graphs, Canad","cited_arxiv_id":null,"evidence_quote":"The f-factor theorem (Lemma 2.4) is the exact criterion that turns absence of a k-factor into the obstruction-family inequality."},{"cited_title":"Zhang, The spectral radius and k-power of Hamilton cycle of graphs, Discrete Math","cited_arxiv_id":null,"evidence_quote":"Supplies the edge-switching lemma (Lemma 2.2) that compares spectral radii of graphs differing by a few edges, the main tool in the extremal maximization."}],"review_version":1}