{"id":"4ca3ddbe-dced-4de2-9d33-721e0c0de761","arxiv_id":"2607.04368","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every fixed ℓ≥2 and large n, H_n is the unique 2n-vertex graph with ≥(n²+n)/2 edges forbidding equal-degree endpoints of a 2ℓ-path.","lead":"The paper proves that the half-graph is the unique densest 2n-vertex graph without equal-degree vertices joined by an even-length path of fixed length 2ℓ. This settles open extremal problems of Chen–Ma and Attwa et al. on degree-constrained paths.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the non-explicit N(ℓ) as the sole material limitation and correctly judges that it does not undermine the asymptotic uniqueness claim. My re-reading of the full proof chain (high-degree control \to equal-degree path lemmas \to multiset sum \to eta-forcing \to half-graph reconstruction) finds no additional load-bearing gap: every numerical constant is tracked, the classical tools (Erdős–Gallai, Naor–Verstraëte) are applied inside their stated ranges, and the final structural identification in Lemma 3.18 is forced once eta = n. Consequently the ACCEPT verdict with high confidence remains appropriate; no adjustment is warranted.","tokens_in":25407,"tokens_out":553,"duration_ms":6254,"concrete_test":"Independently recompute the three quadratic maxima in the proof of Lemma 2.3 (Cases 1–3) with the explicit thresholds α ≥ (2/5)n and the 300ℓ √n cut-off; verify that each maximum is at most n^{2} - 299ℓ n^{3/2} for all n larger than an explicit function of ℓ (e.g., n > 10^6 ℓ^{2}). If the inequality fails for some moderate n, the “sufficiently large” claim needs a concrete N(ℓ); otherwise the asymptotic uniqueness stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central uniqueness claim (Theorem 1.5) is supported by a complete, self-contained argument that proceeds by controlling high-degree vertices (Lemmas 3.1–3.6), equal-degree pairs via core decompositions and path-avoidance (Lemma 3.7), a multiset degree-sum bound (Lemma 2.3), and successive degree-sequence contradictions that force eta = n and the half-graph structure (Lemmas 3.9–3.18). The only non-explicit ingredient is the conventional “sufficiently large n” that absorbs the O(ℓ n^{3/2}) error terms and the numerical thresholds (e.g., 300ℓ √n). This does not create an internal inconsistency or a hidden assumption that would falsify the asymptotic statement; it is the same style of bound used in the odd-length predecessors that the paper cites. No circularity, free parameters, or missing cases appear in the text.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that for every fixed integer ℓ ≥ 2 and all sufficiently large n, the half-graph H_n is the unique graph on 2n vertices with at least (n^{2} + n)/2 edges that contains no two equal-degree vertices joined by a path of length 2ℓ (Theorem 1.5). This determines the extremal function p_{2ℓ}(2n) exactly for large n, answers Problem 1.3 of Chen–Ma, and settles the asymptotic question of Attwa et al. (Problem 1.4). The argument proceeds by controlling the set R of vertices of degree ≥ n + ℓ + 1 (Lemmas 3.1–3.6), analysing equal-degree pairs via a four-layer core decomposition of exclusive neighbourhoods (Lemma 3.7), a multiset degree-sum bound (Lemma 2.3), and a sequence of degree-sequence contradictions that force the maximum equal degree eta to equal n and the graph to be isomorphic to H_n (Lemmas 3.9–3.18).","tokens_in":25648,"tokens_out":957,"duration_ms":9406,"significance":"The result completes the even-length half of the Erdős–Hajnal-type programme initiated by Chen–Ma and continued by Liu–Zeng, Zhao–Wang–Lu and Attwa et al. It supplies a new extremal characterisation of the half-graph, an object of independent interest in model theory and structural graph theory. The proof is fully self-contained, relies only on classical tools (Erdős–Gallai, Naor–Verstraëte), and contains no free parameters or circular reductions. The uniqueness statement is asymptotic, which is the natural level of precision for this series of problems.","major_comments":[],"minor_comments":[{"comment":"The constant 2ℓ + 4 that defines core(H) is never justified beyond the greedy path-construction needs of Lemma 3.7(a). A short remark that any larger constant works and that 2ℓ + 4 is chosen only for convenience would improve readability.","section":"Notation / Lemma 3.7"},{"comment":"Lemma 2.3 is stated for a multiset of 2n integers each at most n; the three cases on α are carefully checked, but the final O(n) terms are absorbed without an explicit lower bound on n. Adding a parenthetical “for n ≥ N(ℓ)” would make the dependence transparent.","section":"Lemma 2.3"},{"comment":"In the definition of the four-layer sets A_{s,i} the intersection with the ambient exclusive neighbourhood is written twice; a single sentence clarifying that the core is taken first and then intersected would remove a minor notational redundancy.","section":"Notation paragraph after D_st"},{"comment":"Several places write “O_ℓ(n^{3/2})” while others write “O(ℓ n^{3/2})”. Uniformising the subscript notation would avoid any momentary confusion.","section":"Throughout §3"},{"comment":"The half-graph is introduced with the classical bipartition {u_i} \times {v_j}, i ≥ j; later the proof reconstructs an isomorphic labelling with parts of size n. A one-line remark that the two presentations differ only by a relabelling would help the reader match the final construction with the definition.","section":"Introduction and Lemma 3.18"}],"recommendation":"accept","confidential_remarks":"The manuscript is a clean, high-quality contribution that sits squarely in the journal’s combinatorial extremal graph theory remit. No novelty or citation concerns arose. The absence of an explicit N(ℓ) is standard for this literature and does not affect the correctness of the asymptotic claim."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes the even-length half of the Chen–Ma program: for every fixed ℓ ≥ 2 and large n, the half-graph H_n is the unique 2n-vertex graph with at least (n² + n)/2 edges that has no equal-degree pair joined by a 2ℓ-path. That settles their Problem 1.3 and the Attwa et al. asymptotic question in one stroke. The odd-length uniqueness theorems and the length-2 case were already known; the new content is the uniqueness statement for even lengths together with a clean structural characterization of H_n.\n\nThe argument is a long chain of lemmas that first control the high-degree set (size O(√n)), then analyze equal-degree pairs via a four-layer core decomposition of exclusive neighborhoods (minimum degree 2ℓ+4), bound common neighborhoods and degree sums with Erdős–Gallai and Naor–Verstraëte, and finally force β = n and the nested neighborhood structure of H_n. The external tools are classical, the numerical thresholds (300ℓ√n etc.) are chosen for convenience, and there is no circularity or free parameter. The full proof text is present and self-contained.\n\nThe only soft spot is the conventional “sufficiently large n” that absorbs the O(ℓ n^{3/2}) error terms; no explicit N(ℓ) is computed. That is the same style used in the odd-length predecessors and does not undermine the asymptotic claim. The core and layer definitions are technical but do the job they are designed for.\n\nThis is for people working on degree-path extremal problems or half-graph structure. It deserves a serious referee. I would accept it for peer review and expect it to go through with only minor polishing.","headline":"Solid resolution of the even-length uniqueness problem for half-graphs; long but complete proof with only the usual asymptotic caveat.","tokens_in":26323,"tokens_out":457,"would_cite":true,"duration_ms":5784,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C38"],"pacs":[],"model":"grok-4.5","headline":"For even-length paths of fixed length, the half graph is the unique densest 2n-vertex graph with no equal-degree endpoints.","keywords":["half graph","equal-degree endpoints","paths of even length","extremal graph theory","degree sequences","Erdős–Hajnal problem"],"falsifier":"Exhibit, for some fixed ℓ ≥ 2 and arbitrarily large n, a 2n-vertex graph with at least (n² + n)/2 edges that is not isomorphic to H_n yet still has no equal-degree pair joined by a 2ℓ-path.","tokens_in":26282,"feed_emoji":"🔀","tokens_out":664,"duration_ms":5245,"temperature":0.7,"pith_summary":"The paper settles a natural even-length counterpart of a classical question of Erdős and Hajnal: how many edges can an n-vertex graph have without two equal-degree vertices joined by a path of prescribed length. Earlier work had completely resolved the odd-length case, with the complete bipartite graph K_{n,n+1} as the unique extremal example. Here the authors prove that for every fixed even length 2ℓ with ℓ ≥ 2 and all sufficiently large n, the unique 2n-vertex graph with at least (n² + n)/2 edges that avoids equal-degree endpoints on a 2ℓ-path is the half graph H_n. The half graph is the bipartite graph on parts of size n in which the i-th vertex of one side is joined to the first i vertices of the other side; it already meets the edge bound and contains no forbidden configuration. The result answers an open problem of Chen and Ma and confirms an asymptotic density conjecture of Attwa et al. for even lengths.","feed_headline":"Half graph is unique densest graph without equal-degree 2ℓ-paths","feed_subtitle":"For every fixed even length and large n, H_n alone meets the (n²+n)/2 edge bound.","key_machinery":"A controlled decomposition of neighborhoods of equal-degree pairs (the sets A_s, A_t, B_st, Y_st and the 2ℓ+4-core of induced subgraphs) together with multiset-sum bounds that force the maximum equal degree β to equal n and the graph to be exactly H_n.","core_discovery":"For every fixed integer ℓ ≥ 2 and all sufficiently large n, the half graph H_n is the unique graph on 2n vertices with at least (n² + n)/2 edges that contains no two vertices of equal degree joined by a path of length 2ℓ.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Half graph unique densest 2n-vertex graph free of equal-degree 2ℓ-paths","H_n sole maximizer of edges without equal-degree paths of length 2ℓ","Unique densest graph avoiding equal-degree even-length paths is half graph","For large n, H_n alone hits (n²+n)/2 edges sans equal-degree 2ℓ-paths","Half graph H_n uniquely achieves edge bound free of equal-degree 2ℓ-paths"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The uniqueness statement holds only for all n larger than some (unspecified) constant depending on ℓ, because the proof relies on asymptotic error terms that dominate only when n is large enough.","fun_headline_variants_meta":{"raw":{"variants":["Half graph unique densest 2n-vertex graph free of equal-degree 2ℓ-paths","H_n sole maximizer of edges without equal-degree paths of length 2ℓ","Unique densest graph avoiding equal-degree even-length paths is half graph","For large n, H_n alone hits (n²+n)/2 edges sans equal-degree 2ℓ-paths","Half graph H_n uniquely achieves edge bound free of equal-degree 2ℓ-paths"]},"model":"grok-4.5","effort":"low","cost_usd":0.005546,"raw_usage":{"total_tokens":1441,"prompt_tokens":727,"num_sources_used":0,"completion_tokens":127,"cost_in_usd_ticks":55460000,"prompt_tokens_details":{"text_tokens":727,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":587,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":727,"tokens_out":127,"duration_ms":5332,"temperature":1.0,"reasoning_tokens":587,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T19:43:00.487866+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit, for some fixed ℓ ≥ 2 and arbitrarily large n, a 2n-vertex graph with at least (n² + n)/2 edges that is not isomorphic to H_n yet still has no equal-degree pair joined by a 2ℓ-path.","supporting_citations":[],"review_version":1}