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A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible.\n  Let ${\\bf v} \\in {\\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $\\nu_{\\bf v}(L) = \\sum_{F \\in F_k} {\\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1902.00682","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-02-02T10:14:35Z","cross_cats_sorted":[],"title_canon_sha256":"285f5762d74fb10b3cd8f857ece1283d115338d79b6baa5c39ef75063d33af44","abstract_canon_sha256":"96794e955e80b690613b92003a349c9e965bd8ede9165b322c80c0e80098e48c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:52.447447Z","signature_b64":"H08qHM5sf3uTrjTbpp5UBsSzlBhNlzu3mlb4/ip3BmtykN9hhxtID/P5qLY3FbDxWEIicPitZVGD76SWm+yQDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38cad356f47d4537ffb210e0c3d61c67fdf779f98de77f2699b9eac9baf32fcb","last_reissued_at":"2026-05-17T23:54:52.446927Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:52.446927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vector clique decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Raphael Yuster","submitted_at":"2019-02-02T10:14:35Z","abstract_excerpt":"Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set. A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible.\n  Let ${\\bf v} \\in {\\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $\\nu_{\\bf v}(L) = \\sum_{F \\in F_k} {\\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00682","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1902.00682","created_at":"2026-05-17T23:54:52.447013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.00682v1","created_at":"2026-05-17T23:54:52.447013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.00682","created_at":"2026-05-17T23:54:52.447013+00:00"},{"alias_kind":"pith_short_12","alias_value":"HDFNGVXUPVCT","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"HDFNGVXUPVCTP75S","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"HDFNGVXU","created_at":"2026-05-18T12:33:18.533446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7","json":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7.json","graph_json":"https://pith.science/api/pith-number/HDFNGVXUPVCTP75SCDQMHVQ4M7/graph.json","events_json":"https://pith.science/api/pith-number/HDFNGVXUPVCTP75SCDQMHVQ4M7/events.json","paper":"https://pith.science/paper/HDFNGVXU"},"agent_actions":{"view_html":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7","download_json":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7.json","view_paper":"https://pith.science/paper/HDFNGVXU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.00682&json=true","fetch_graph":"https://pith.science/api/pith-number/HDFNGVXUPVCTP75SCDQMHVQ4M7/graph.json","fetch_events":"https://pith.science/api/pith-number/HDFNGVXUPVCTP75SCDQMHVQ4M7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7/action/storage_attestation","attest_author":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7/action/author_attestation","sign_citation":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7/action/citation_signature","submit_replication":"https://pith.science/pith/HDFNGVXUPVCTP75SCDQMHVQ4M7/action/replication_record"}},"created_at":"2026-05-17T23:54:52.447013+00:00","updated_at":"2026-05-17T23:54:52.447013+00:00"}