{"paper":{"title":"Induced subdivisions in graphs of large girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Graphs with minimum degree at least k and girth above a fixed constant contain an induced subdivision of K_{k+1}.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peiru Kuang, Yan Wang","submitted_at":"2026-05-17T01:32:55Z","abstract_excerpt":"In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\\ge 3$, every graph $G$ with $\\delta(G)\\ge k$ and $g(G)\\ge g_0$ contains an induced subdivision of $K_{k+1}$. This answers, in a strong sense, a problem asked by K\\\"uhn and Osthus (originally attributed to Shi). A main ingredient in our proof is an induced variant of Mader's theorem: for every fixed \\(s,\\eta,D\\), every graph \\(J\\) with \\(\\Delta(J)\\le D\\), \\(d(J)>s-2+\\eta\\) and sufficiently large girth contains an induced subdivision of \\(K_s\\)."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"there exists an absolute constant g0 such that, for every integer k≥3, every graph G with δ(G)≥k and g(G)≥g0 contains an induced subdivision of K_{k+1}","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof depends on an induced variant of Mader's theorem (for every fixed s, η, D, every graph J with Δ(J)≤D, d(J)>s−2+η and sufficiently large girth contains an induced subdivision of K_s) whose own proof is not supplied in the abstract and is treated as a black-box ingredient.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"There exists an absolute constant g0 such that every graph with minimum degree at least k and girth at least g0 contains an induced subdivision of K_{k+1}.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Graphs with minimum degree at least k and girth above a fixed constant contain an induced subdivision of K_{k+1}.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0d616829ca71eef9d52ea29b3a1573c22c1974389f9a2a2ef9d96cb5105642a4"},"source":{"id":"2605.17218","kind":"arxiv","version":1},"verdict":{"id":"d91ecf00-84cb-40ec-85cd-1597ef35b5ba","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:35:39.186918Z","strongest_claim":"there exists an absolute constant g0 such that, for every integer k≥3, every graph G with δ(G)≥k and g(G)≥g0 contains an induced subdivision of K_{k+1}","one_line_summary":"There exists an absolute constant g0 such that every graph with minimum degree at least k and girth at least g0 contains an induced subdivision of K_{k+1}.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proof depends on an induced variant of Mader's theorem (for every fixed s, η, D, every graph J with Δ(J)≤D, d(J)>s−2+η and sufficiently large girth contains an induced subdivision of K_s) whose own proof is not supplied in the abstract and is treated as a black-box ingredient.","pith_extraction_headline":"Graphs with minimum degree at least k and girth above a fixed constant contain an induced subdivision of K_{k+1}."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17218/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:20.686340Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:52:41.582939Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.715995Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.921650Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"beef8ec048002a99cbd68930251f10766127e125ac16119556757c869b462964"},"references":{"count":36,"sample":[{"doi":"","year":2002,"title":"N. 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