{"paper":{"title":"Fractional clique decompositions of dense balanced multipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Balanced r-partite graphs with high partite minimum degree admit fractional K_s-decompositions for small enough density deficits depending on r and s.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hengrui Liu, Shikang Yu, Tao Feng","submitted_at":"2026-04-28T04:23:55Z","abstract_excerpt":"This paper concerns fractional $K_s$-decompositions of multipartite graphs. For integers $r\\ge s\\ge 3$, we consider balanced $r$-partite graphs $G$ on $rn$ vertices. We establish necessary conditions for $G$ to admit a fractional $K_s$-decomposition, extending the notion of $s$-admissibility from the case $r=s$ to $r>s$. Using an association scheme on the edge set of a complete $r$-partite graph, we prove that if $r\\ge s+2$ and the partite minimum degree of $G$ is at least $(1-c)n$ with $c\\le 1/((s-2)(s+1)(s-1)^4)$, then $G$ has a fractional $K_s$-decomposition. For $r=s+1$, we show that under"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if r≥s+2 and the partite minimum degree of G is at least (1-c)n with c≤1/((s-2)(s+1)(s-1)^4), then G has a fractional K_s-decomposition; for r=s+1 the bound is c≤1/(3s^3(s-2)^2) under s-admissibility.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graphs must be balanced (equal part sizes) and the s-admissibility condition must hold; the association scheme averaging argument assumes the density is high enough for the error terms to be controlled by the given c bounds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Balanced r-partite graphs with partite minimum degree at least (1-c)n admit fractional K_s-decompositions for r >= s+1 under explicit c bounds that depend on s and the gap between r and s.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Balanced r-partite graphs with high partite minimum degree admit fractional K_s-decompositions for small enough density deficits depending on r and s.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e9a77685293f85537249f607d162a9d9adee163443837946aa685ef1343ddd40"},"source":{"id":"2604.25206","kind":"arxiv","version":2},"verdict":{"id":"8fd16ff8-c56c-423f-9e90-09dd4a32fb87","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T16:00:51.545802Z","strongest_claim":"if r≥s+2 and the partite minimum degree of G is at least (1-c)n with c≤1/((s-2)(s+1)(s-1)^4), then G has a fractional K_s-decomposition; for r=s+1 the bound is c≤1/(3s^3(s-2)^2) under s-admissibility.","one_line_summary":"Balanced r-partite graphs with partite minimum degree at least (1-c)n admit fractional K_s-decompositions for r >= s+1 under explicit c bounds that depend on s and the gap between r and s.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graphs must be balanced (equal part sizes) and the s-admissibility condition must hold; the association scheme averaging argument assumes the density is high enough for the error terms to be controlled by the given c bounds.","pith_extraction_headline":"Balanced r-partite graphs with high partite minimum degree admit fractional K_s-decompositions for small enough density deficits depending on r and s."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25206/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T05:35:10.977157Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:23:27.553405Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d4645e3df1480acde721b1c64ae938d12b06329e29675776835f35a6905d4fec"},"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"}