{"id":"947b94a4-4354-47b0-a5f2-96d4f0f19381","arxiv_id":"2607.04323","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every n-vertex graph with odd girth at least 2k+1 and minimum degree greater than 4n/(6k−1) is homomorphic to the Möbius ladder M_{4k}.","lead":"Dense graphs without short odd cycles must be homomorphic to a Möbius ladder once minimum degree exceeds 4n/(6k−1). The result tightens classical Andrásfai–Erdős–Sós structure theorems and answers an open question of Messuti and Schacht.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates maximality as the technical engine that supplies the controlled-length paths used to build the forbidden configurations. That engine is classical (AES-style) and is applied carefully: the proofs repeatedly invoke the odd-girth hypothesis to rule out shorter odd closed walks, thereby forcing length exactly 2k-2 and interior-disjointness. The remainder of the argument (Claims 1-18, degree double-counting on the resulting 6k-1-vertex subgraphs, and the final blow-up maximality) is a long but standard nested case analysis with no free parameters. Residual risk is only the ordinary possibility of a missed subcase in a combinatorial proof of this length; no concrete inconsistency or hidden assumption is visible. Hence the ACCEPT verdict stands.","tokens_in":32236,"tokens_out":489,"duration_ms":6120,"concrete_test":"Independently re-derive the length-control step of Lemma 3.4 (and the analogous step in Lemma 4.1) for the base cases k=2 and k=3: verify that every missing diagonal of an induced C6,1 is completed by a path of length exactly 2k-2 whose interior is disjoint from the C6,1 and that the two such paths are vertex-disjoint; if either length or disjointness fails for a concrete maximal graph on 6k-1 vertices, the configuration-expansion chain collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The maximality assumption flagged by the reader is standard and correctly applied: for a maximal {C3,...,C2k-1}-free graph, any non-edge is completed by an even path of length at most 2k-2, and the proofs of Lemmas 3.4 and 4.1 carefully force the length to be exactly 2k-2 by forbidding shorter odd closed walks. The subsequent configuration expansions (diagonal Φ4k,1 \to Φ4k,3 \to M4k; T1 tetrahedra \to length-one spoke \to M4k) and the final blow-up maximality argument in §5 are self-contained and use only the odd-girth hypothesis plus degree counting. No concrete gap or uncontrolled case appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that every n-vertex graph G with odd girth at least 2k+1 and minimum degree δ(G) > 4n/(6k-1) is homomorphic to the Möbius ladder M_{4k}. This strengthens the Andrásfai–Erdős–Sós theorem (homomorphism to K_2 under the weaker bound 2n/(2k+1)) and the Messuti–Schacht theorem (homomorphism to C_{2k+1} under the bound 3n/(4k)), answers a question of Messuti and Schacht, and generalizes the Brandt–Ribe-Baumann result for girth 7. The argument proceeds by maximality reduction to the class G_{n,k}, configuration expansion via diagonal Φ_{4k,1}/Φ_{4k,3} (Lemmas 3.1–3.2) and (2k+1)-tetrahedra in T_1 (Lemmas 4.3–4.4), and a final absorption argument showing that any maximal blow-up of M_{4k} must be the whole graph (Section 5, using forbidden induced Ψ_1/Ψ_2 and degree counting).","tokens_in":32477,"tokens_out":850,"duration_ms":8965,"significance":"The result sits cleanly in the classical line of Andrásfai–Erdős–Sós, Häggkvist, Jin, Brandt–Ribe-Baumann, Messuti–Schacht and Letzter–Snyder: it supplies the next natural host graph (M_{4k}) under a degree threshold that the authors and Messuti–Schacht already conjectured to be optimal via blow-ups of a (6k-1)-cycle with selected chords. The proof is a self-contained combinatorial case analysis that does not rely on regularity or probabilistic methods; the degree bound is an external hypothesis, the host is a fixed finite graph, and the maximality reduction is standard. The work therefore advances the structural theory of dense graphs of large odd girth and settles an explicit open question.","major_comments":[],"minor_comments":[{"comment":"In the definition of F_{ℓ,k} (page 1) the phrase “distance of the form j(2k-1)+1” is slightly ambiguous; a short clarifying sentence or a reference to the standard generalized Andrásfai graph would help.","section":null},{"comment":"Lemma 2.1 and Lemma 2.2 are used repeatedly; a one-sentence reminder of their conclusions at the first major application (e.g., in the proof of Lemma 3.1) would improve readability.","section":null},{"comment":"Figure 1 labels M_{4k} and Φ_{4k,3}; the same style of labelling for the tetrahedron configurations in Section 4 would make the case distinctions easier to follow.","section":null},{"comment":"In Section 5 the graphs Ψ_1 and Ψ_2 are introduced without a forward reference; a brief sentence explaining that they are the only remaining forbidden configurations needed for the blow-up maximality argument would clarify the logical flow.","section":null},{"comment":"A few minor typos appear (e.g., “aÿirmative”, “consequenty”, occasional missing articles); a careful copy-edit pass is recommended.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, self-contained combinatorial paper that answers a concrete question of Messuti and Schacht. The case analysis is long but appears complete; I found no load-bearing gap. Suitable for a combinatorics journal of good standing."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the natural next step after Messuti–Schacht: under odd girth at least 2k+1, δ>4n/(6k−1) already forces a homomorphism to the Möbius ladder M_{4k}. That bound is exactly the one they conjectured, and the blow-ups of the (6k−1)-cycle with every 2k-chord give the matching extremal examples. It also recovers Brandt–Ribe-Baumann for k=3 as a special case.\n\nWhat they do well is keep the classical AES skeleton (maximal free graphs, degree counting, configuration expansion) and make it work for general k. The two new gadgets—diagonal Φ_{4k,1}/Φ_{4k,3} and the (2k+1)-tetrahedra T_1—are cleanly defined, and Lemmas 3.1–3.2 and 4.3–4.4 show that each of them forces an induced M_{4k}. Section 5 then absorbs every leftover vertex into a maximal blow-up by forbidding two small Ψ graphs and a short double-counting argument. The maximality reduction is the usual one and is applied carefully: non-edges produce even paths of length exactly 2k−2, not merely “at most.” No free parameters, no circular fits.\n\nThe soft spot is the length of the nested case analysis. There are many sub-claims about neighbor locations and path parities; a missed configuration is always possible in a proof of this style. But the stress-test found none, and the counting steps themselves are tight and elementary. The imported Lemma 5.2 from Messuti–Schacht is used only as a clean base case, not as a black box that hides work.\n\nThis is for people already inside the homomorphism-threshold / Andrásfai–Erdős–Sós line. If you care about the next degree threshold after C_{2k+1}, you will want the statement and the extremal construction. It does not invent new technology, but it is a correctly scoped, self-contained combinatorial theorem. I would send it to a serious referee without hesitation; the residual risk is ordinary case-checking, not conceptual.\n\nRecommendation: accept for peer review.","headline":"Solid resolution of Messuti–Schacht’s question: the 4n/(6k−1) threshold forces Hom(G,M_{4k}) for odd girth ≥2k+1, with matching extremal construction.","tokens_in":33101,"tokens_out":582,"would_cite":true,"duration_ms":7423,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","05C38","05C35"],"pacs":[],"model":"grok-4.5","headline":"Dense graphs with large odd girth must map homomorphically onto a Möbius ladder.","keywords":["homomorphism","odd girth","Möbius ladder","minimum degree","graph homomorphism threshold","maximal graphs"],"falsifier":"Exhibit a single n-vertex graph of odd girth at least 2k+1, minimum degree greater than 4n/(6k-1), that admits no homomorphism into M_{4k} (for example a suitable blow-up of a tetrahedron or of the conjectured extremal (6k-1)-cycle with chords).","tokens_in":33132,"feed_emoji":"🔄","tokens_out":988,"duration_ms":17889,"temperature":0.7,"pith_summary":"The paper proves that any n-vertex graph whose shortest odd cycle is at least 2k+1 long and whose minimum degree exceeds 4n/(6k-1) admits a homomorphism into the Möbius ladder on 4k vertices. Earlier density thresholds already forced such graphs to be bipartite or to map onto an odd cycle; the new bound sits between those thresholds and forces a richer but still fixed target. The result settles an open question about whether the next natural target after the cycle is the Möbius ladder, and it recovers known special cases for small girth. A sympathetic reader cares because the statement gives a precise structural description of all sufficiently dense graphs that avoid short odd cycles: they are essentially blow-ups of one fixed ladder graph.","feed_headline":"Dense odd-girth graphs map onto a Möbius ladder","feed_subtitle":"Degree above 4n/(6k-1) forces a homomorphism to the 4k-vertex ladder, settling a structural question","key_machinery":"The argument expands two families of forbidden configurations—diagonal Φ_{4k,1}/Φ_{4k,3} subgraphs and (2k+1)-tetrahedra with long spokes—until each forces an induced copy of M_{4k}; maximality then lets missing edges be completed by controlled even paths that generate these configurations.","core_discovery":"For every integer k at least 2, every n-vertex graph of odd girth at least 2k+1 and minimum degree strictly larger than 4n/(6k-1) is homomorphic to the Möbius ladder M_{4k} obtained from a 4k-cycle by adding all diameters. The same degree bound is conjectured to be optimal, witnessed by certain blow-ups of (6k-1)-cycles with selected chords.","pith_inferences":["If the conjectured extremal blow-ups of chordal (6k-1)-cycles can be shown to have homomorphism number strictly larger than that of M_{4k}, the degree threshold is sharp.","The same configuration-expansion technique may push the threshold lower still, toward the open question of whether degree n/(2k-2) already forces a homomorphism into some generalised Andrásfai graph F_{ℓ,k}.","A computer search for small-k counter-examples just below the bound would quickly test whether maximality can be relaxed."],"forward_implications":["Any graph meeting the degree and girth hypotheses is 3-colourable, since M_{4k} is 3-colourable.","The earlier cycle-homomorphism threshold is improved: the same graphs map onto the richer fixed target M_{4k} rather than merely C_{2k+1}.","When the minimum degree exceeds 4n/(6k-1) the only maximal examples are blow-ups of M_{4k} itself.","The bound specialises for k=2 and k=3 to previously known statements about triangle-free and {C3,C5}-free graphs."],"fun_headline_variants":["Odd-girth graphs with δ>4n/(6k-1) map to Möbius ladder M_4k","High min-degree odd-girth graphs are homomorphic to M_4k","δ>4n/(6k-1) forces odd-girth≥2k+1 graphs onto 4k Möbius ladder","Dense graphs of odd girth map to the Möbius ladder on 4k verts","Odd girth and degree bound yield Möbius-ladder homomorphisms"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The graphs are assumed maximal: every missing edge can be completed by an even path of length exactly 2k-2 that creates a (2k+1)-cycle, and the whole proof relies on the existence of those paths.","fun_headline_variants_meta":{"raw":{"variants":["Odd-girth graphs with δ>4n/(6k-1) map to Möbius ladder M_4k","High min-degree odd-girth graphs are homomorphic to M_4k","δ>4n/(6k-1) forces odd-girth≥2k+1 graphs onto 4k Möbius ladder","Dense graphs of odd girth map to the Möbius ladder on 4k verts","Odd girth and degree bound yield Möbius-ladder homomorphisms"]},"model":"grok-4.5","effort":"low","cost_usd":0.008776,"raw_usage":{"total_tokens":2038,"prompt_tokens":767,"num_sources_used":0,"completion_tokens":132,"cost_in_usd_ticks":87760000,"prompt_tokens_details":{"text_tokens":767,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1139,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":767,"tokens_out":132,"duration_ms":9836,"temperature":1.0,"reasoning_tokens":1139,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T20:04:10.623551+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single n-vertex graph of odd girth at least 2k+1, minimum degree greater than 4n/(6k-1), that admits no homomorphism into M_{4k} (for example a suitable blow-up of a tetrahedron or of the conjectured extremal (6k-1)-cycle with chords).","supporting_citations":[],"review_version":1}