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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","authors_text":"Hengrui Liu, Shikang Yu, Tao Feng","cross_cats":[],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-28T04:23:55Z","title":"Fractional clique decompositions of dense balanced multipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25206","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T16:00:51.545802Z","id":"8fd16ff8-c56c-423f-9e90-09dd4a32fb87","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"8fd16ff8-c56c-423f-9e90-09dd4a32fb87"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b86443c946a0e57837235edf58d8b1d0e494bab8e7d176552a76cd390b79ab86","target":"record","created_at":"2026-06-25T01:17:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"57f3d951c4aa8f707db15bdbcb504ba99b80a415e46fcde4992e4884a2ef6dd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-04-28T04:23:55Z","title_canon_sha256":"35d481cb58e797e38ccc889f1e2dd96f0936946d0e9be3db59bdfbbcaf916bb5"},"schema_version":"1.0","source":{"id":"2604.25206","kind":"arxiv","version":2}},"canonical_sha256":"18157e3cf396de8edac53194d56f804fff2fb2e55699a92056a1c9939723e390","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18157e3cf396de8edac53194d56f804fff2fb2e55699a92056a1c9939723e390","first_computed_at":"2026-06-25T01:17:53.503485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-25T01:17:53.503485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zcRB6kutv+lW6GeM/eobU6A0BNBa/AbuZ/aDIQ+tLsGkG/rdOP+sYV7zDrzUwFG/m9azf7dWJ3ArCsbhSpzWBg==","signature_status":"signed_v1","signed_at":"2026-06-25T01:17:53.503914Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.25206","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b86443c946a0e57837235edf58d8b1d0e494bab8e7d176552a76cd390b79ab86","sha256:d357b1799fdb0da0298c411e972fed9092c434d724dea5944b4b281026f720be"],"state_sha256":"45b1eaee336ab4f0a1370c84df9d679d7d2caf2849a078dc91e59de74a4ca408"}