{"paper":{"title":"Wheel Classes in Kontsevich Graph Complex and Merkulov's Low-Valence Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Wheel graphs in the Kontsevich graph complex are homologous to explicit combinations of only 3- and 4-valent graphs.","cross_cats":["math.CO"],"primary_cat":"math.QA","authors_text":"Assar Andersson","submitted_at":"2026-04-13T11:29:44Z","abstract_excerpt":"We show that the wheel classes in the Kontsevich graph complex $GC_d$ admit representatives supported on graphs with only $3$- and $4$-valent vertices. This verifies that Merkulov's low-valence conjecture holds for the wheel classes.\n  More precisely, for every $m \\ge 2$, we prove that the wheel graph $W_{2m+1}$ is homologous to an explicit linear combination of $2^{m-2}$ graphs, each having only $3$- and $4$-valent vertices."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for every m ≥ 2, we prove that the wheel graph W_{2m+1} is homologous to an explicit linear combination of 2^{m-1} graphs, each having only 3- and 4-valent vertices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The explicit linear combination constructed in the paper is a cycle whose boundary equals that of the wheel graph under the standard differential of GC_d.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For every m ≥ 2 the wheel graph W_{2m+1} is homologous to an explicit sum of 2^{m-1} graphs with only 3- and 4-valent vertices.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Wheel graphs in the Kontsevich graph complex are homologous to explicit combinations of only 3- and 4-valent graphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"49597b15c98569a544763d3bff1efff60df38650d1af5483a0fb35af8c9d2745"},"source":{"id":"2604.11327","kind":"arxiv","version":2},"verdict":{"id":"95f7cf7c-c132-4e11-947f-09ba0c4eb57f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:40:51.261057Z","strongest_claim":"for every m ≥ 2, we prove that the wheel graph W_{2m+1} is homologous to an explicit linear combination of 2^{m-1} graphs, each having only 3- and 4-valent vertices.","one_line_summary":"For every m ≥ 2 the wheel graph W_{2m+1} is homologous to an explicit sum of 2^{m-1} graphs with only 3- and 4-valent vertices.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The explicit linear combination constructed in the paper is a cycle whose boundary equals that of the wheel graph under the standard differential of GC_d.","pith_extraction_headline":"Wheel graphs in the Kontsevich graph complex are homologous to explicit combinations of only 3- and 4-valent graphs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11327/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}