{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:TAAJFDHB3W5TWMMXBWHHJOZBWW","short_pith_number":"pith:TAAJFDHB","schema_version":"1.0","canonical_sha256":"9800928ce1ddbb3b31970d8e74bb21b58a0b7cff80817fe818216f07e9f0fb52","source":{"kind":"arxiv","id":"1809.01443","version":1},"attestation_state":"computed","paper":{"title":"On Clique Coverings of Complete Multipartite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Akbar Davoodi, D\\'aniel Gerbner, M\\'at\\'e Vizer","submitted_at":"2018-09-05T11:37:49Z","abstract_excerpt":"A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number $scc(G)$ of a graph $G$, is defined as the smallest possible weight of a clique covering of $G$. Let $ K_t(d) $ denote the complete $ t $-partite graph with each part of size $d$. We prove that for any fixed $d \\ge 2$, we have $$\\lim_{t \\rightarrow \\infty} scc(K_t(d))= \\frac{d}{2} t\\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi."},"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":"1809.01443","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-05T11:37:49Z","cross_cats_sorted":[],"title_canon_sha256":"02fb2d3d55c147b3755360e9b4cd6261c4d7811a0171ad74155f34329c09391e","abstract_canon_sha256":"6df5baab4260f9bd9e9ae14b0975f34867871f781a7c3914e939632eaa1f974b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:27.650866Z","signature_b64":"XkRaEx7S+LA6PmRf+OOhbhyAGLHmaqEEN0CmdICi7QzF9yCUknNk+OX494Dw88XrCrCswIQpMVwmJNLnm3UoCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9800928ce1ddbb3b31970d8e74bb21b58a0b7cff80817fe818216f07e9f0fb52","last_reissued_at":"2026-05-18T00:06:27.650179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:27.650179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Clique Coverings of Complete Multipartite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Akbar Davoodi, D\\'aniel Gerbner, M\\'at\\'e Vizer","submitted_at":"2018-09-05T11:37:49Z","abstract_excerpt":"A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number $scc(G)$ of a graph $G$, is defined as the smallest possible weight of a clique covering of $G$. Let $ K_t(d) $ denote the complete $ t $-partite graph with each part of size $d$. We prove that for any fixed $d \\ge 2$, we have $$\\lim_{t \\rightarrow \\infty} scc(K_t(d))= \\frac{d}{2} t\\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01443","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":"1809.01443","created_at":"2026-05-18T00:06:27.650284+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.01443v1","created_at":"2026-05-18T00:06:27.650284+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.01443","created_at":"2026-05-18T00:06:27.650284+00:00"},{"alias_kind":"pith_short_12","alias_value":"TAAJFDHB3W5T","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"TAAJFDHB3W5TWMMX","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"TAAJFDHB","created_at":"2026-05-18T12:32:53.628368+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/TAAJFDHB3W5TWMMXBWHHJOZBWW","json":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW.json","graph_json":"https://pith.science/api/pith-number/TAAJFDHB3W5TWMMXBWHHJOZBWW/graph.json","events_json":"https://pith.science/api/pith-number/TAAJFDHB3W5TWMMXBWHHJOZBWW/events.json","paper":"https://pith.science/paper/TAAJFDHB"},"agent_actions":{"view_html":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW","download_json":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW.json","view_paper":"https://pith.science/paper/TAAJFDHB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.01443&json=true","fetch_graph":"https://pith.science/api/pith-number/TAAJFDHB3W5TWMMXBWHHJOZBWW/graph.json","fetch_events":"https://pith.science/api/pith-number/TAAJFDHB3W5TWMMXBWHHJOZBWW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW/action/storage_attestation","attest_author":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW/action/author_attestation","sign_citation":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW/action/citation_signature","submit_replication":"https://pith.science/pith/TAAJFDHB3W5TWMMXBWHHJOZBWW/action/replication_record"}},"created_at":"2026-05-18T00:06:27.650284+00:00","updated_at":"2026-05-18T00:06:27.650284+00:00"}