{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7PRSDBOGGWNDQWL3MV2FLXV4AH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"15193e17166e2fc41883d97526b71b81c7366a752206dd50e9c937ee79b7f7f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-06T21:24:48Z","title_canon_sha256":"3ebb5ae1ba94a070d21887bf0b93d3fc32a3fb23417eb36a5be3537f6731f0fe"},"schema_version":"1.0","source":{"id":"1203.1336","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1336","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1336v1","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1336","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"pith_short_12","alias_value":"7PRSDBOGGWND","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7PRSDBOGGWNDQWL3","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7PRSDBOG","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:9b2ae6410e77e88c9dc064c7bd5ad050129a470a9bf6983d87e45cd2ed756f41","target":"graph","created_at":"2026-05-18T04:00:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct extremal graphs that achieve the edge bounds in all cases. We further establish uniqueness of these extremal graphs whenever they are unique.","authors_text":"Naushad Puliyambalath, Niraj Khare, Nishali Mehta","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-06T21:24:48Z","title":"Maximum edges possible in a graph for restricted independence number, maximum degree, and maximum matching size"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1336","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1fa21af988321b8f285956a0612260acee9d3b8bcd8d3bda4ec5bfffa734b9da","target":"record","created_at":"2026-05-18T04:00:39Z","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":"15193e17166e2fc41883d97526b71b81c7366a752206dd50e9c937ee79b7f7f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-06T21:24:48Z","title_canon_sha256":"3ebb5ae1ba94a070d21887bf0b93d3fc32a3fb23417eb36a5be3537f6731f0fe"},"schema_version":"1.0","source":{"id":"1203.1336","kind":"arxiv","version":1}},"canonical_sha256":"fbe32185c6359a38597b657455debc01f53f1bff95586bd1202008fe6b2d0d87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbe32185c6359a38597b657455debc01f53f1bff95586bd1202008fe6b2d0d87","first_computed_at":"2026-05-18T04:00:39.644253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:39.644253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Eo1qabHDxopo2HsWqpucuRjwGFa3SP0l5g69hqGour1AsEyfFZZ9dFP9D3tLWBI+QCVtyHezUQP3OM8w+zGZCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:39.644743Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.1336","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1fa21af988321b8f285956a0612260acee9d3b8bcd8d3bda4ec5bfffa734b9da","sha256:9b2ae6410e77e88c9dc064c7bd5ad050129a470a9bf6983d87e45cd2ed756f41"],"state_sha256":"3eaf287f2102a8913c6582945642822240f0943eb34a940bb2a730769d97c772"}