{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:GWJF4DTAORZODOLJVQIQAY6FXE","short_pith_number":"pith:GWJF4DTA","schema_version":"1.0","canonical_sha256":"35925e0e607472e1b969ac110063c5b9018aa6708ea11fa39b3e1d969f9546ab","source":{"kind":"arxiv","id":"2606.08506","version":1},"attestation_state":"computed","paper":{"title":"Almost balanced ordered biclique covering of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Anagh Indu Suresh, Anand Babu, Ervin Ranjan, Jatla Naga Sidhartha, Maddipati Deshith Sai, Sreedhara Vishwas","submitted_at":"2026-06-07T08:14:03Z","abstract_excerpt":"Let $f(n,k)$ be the minimum size of a collection of bicliques such that (i) every edge of the complete graph $K_n$ is covered by at least one and at most $k$ bicliques in the collection, and (ii) for each edge $\\{u,v\\}$, the number of bicliques in which $u$ appears in the first class and $v$ in the second class differs by at most one from the number of bicliques in which $u$ appears in the second class and $v$ in the first class.\n  For $k=1$, $f(n,k)$ reduces to the biclique partition number of $K_n$, and the Graham--Pollak theorem gives $f(n,1)=n-1$. For $k=2$, $f(n,k)$ is the ordered bicliqu"},"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":"2606.08506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-07T08:14:03Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3c25914410922beb58a07022f76e3fd2c449feabaf5f838b796bc74367e75d22","abstract_canon_sha256":"e896aa1af12640e80dce846622b9ed30fb68c3f8c20fea29c5aa56cea77ee045"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:38.547755Z","signature_b64":"H372VeArrBT4O4iHylhzsRWTXe5jQ9I0KffkymLfZCgNJqaUOEgcIUxcbIUdxN41eQWsCq5AHWFuocUaebp1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35925e0e607472e1b969ac110063c5b9018aa6708ea11fa39b3e1d969f9546ab","last_reissued_at":"2026-06-09T01:05:38.547338Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:38.547338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost balanced ordered biclique covering of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Anagh Indu Suresh, Anand Babu, Ervin Ranjan, Jatla Naga Sidhartha, Maddipati Deshith Sai, Sreedhara Vishwas","submitted_at":"2026-06-07T08:14:03Z","abstract_excerpt":"Let $f(n,k)$ be the minimum size of a collection of bicliques such that (i) every edge of the complete graph $K_n$ is covered by at least one and at most $k$ bicliques in the collection, and (ii) for each edge $\\{u,v\\}$, the number of bicliques in which $u$ appears in the first class and $v$ in the second class differs by at most one from the number of bicliques in which $u$ appears in the second class and $v$ in the first class.\n  For $k=1$, $f(n,k)$ reduces to the biclique partition number of $K_n$, and the Graham--Pollak theorem gives $f(n,1)=n-1$. For $k=2$, $f(n,k)$ is the ordered bicliqu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08506","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08506/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.08506","created_at":"2026-06-09T01:05:38.547401+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08506v1","created_at":"2026-06-09T01:05:38.547401+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08506","created_at":"2026-06-09T01:05:38.547401+00:00"},{"alias_kind":"pith_short_12","alias_value":"GWJF4DTAORZO","created_at":"2026-06-09T01:05:38.547401+00:00"},{"alias_kind":"pith_short_16","alias_value":"GWJF4DTAORZODOLJ","created_at":"2026-06-09T01:05:38.547401+00:00"},{"alias_kind":"pith_short_8","alias_value":"GWJF4DTA","created_at":"2026-06-09T01:05:38.547401+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/GWJF4DTAORZODOLJVQIQAY6FXE","json":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE.json","graph_json":"https://pith.science/api/pith-number/GWJF4DTAORZODOLJVQIQAY6FXE/graph.json","events_json":"https://pith.science/api/pith-number/GWJF4DTAORZODOLJVQIQAY6FXE/events.json","paper":"https://pith.science/paper/GWJF4DTA"},"agent_actions":{"view_html":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE","download_json":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE.json","view_paper":"https://pith.science/paper/GWJF4DTA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08506&json=true","fetch_graph":"https://pith.science/api/pith-number/GWJF4DTAORZODOLJVQIQAY6FXE/graph.json","fetch_events":"https://pith.science/api/pith-number/GWJF4DTAORZODOLJVQIQAY6FXE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE/action/storage_attestation","attest_author":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE/action/author_attestation","sign_citation":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE/action/citation_signature","submit_replication":"https://pith.science/pith/GWJF4DTAORZODOLJVQIQAY6FXE/action/replication_record"}},"created_at":"2026-06-09T01:05:38.547401+00:00","updated_at":"2026-06-09T01:05:38.547401+00:00"}