{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZMTIEGG6B7ZI3JSL72VEBJG6FO","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":"b9878792d55cc29a55ee352349d9ae27b94bafd25e478ad3c2ab921fbb1eb315","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-26T08:05:58Z","title_canon_sha256":"79564997d1df6af104e19ee0d2f2feec599bb1522432fb1a83d466b243a8f170"},"schema_version":"1.0","source":{"id":"1105.5225","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.5225","created_at":"2026-05-18T04:03:43Z"},{"alias_kind":"arxiv_version","alias_value":"1105.5225v2","created_at":"2026-05-18T04:03:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5225","created_at":"2026-05-18T04:03:43Z"},{"alias_kind":"pith_short_12","alias_value":"ZMTIEGG6B7ZI","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZMTIEGG6B7ZI3JSL","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZMTIEGG6","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:70f8ea80c08c2df099add0d4b1563faf4b62916282b207ec16f1a2455e316aca","target":"graph","created_at":"2026-05-18T04:03:43Z","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":"A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\\times R_2\\times ...\\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes.\n  It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the inter","authors_text":"Abhijin Adiga, L. Sunil Chandran, Rogers Mathew","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-26T08:05:58Z","title":"Cubicity, Degeneracy, and Crossing Number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5225","kind":"arxiv","version":2},"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":"sha256:954dff6b6dd7f6ba7ee4cd1f607640bebd9cd47db39e900d74f3e6bdf19ba932","target":"record","created_at":"2026-05-18T04:03:43Z","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":"b9878792d55cc29a55ee352349d9ae27b94bafd25e478ad3c2ab921fbb1eb315","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-26T08:05:58Z","title_canon_sha256":"79564997d1df6af104e19ee0d2f2feec599bb1522432fb1a83d466b243a8f170"},"schema_version":"1.0","source":{"id":"1105.5225","kind":"arxiv","version":2}},"canonical_sha256":"cb268218de0ff28da64bfeaa40a4de2bae03b584a0aab46f9fe2187d5e27be94","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb268218de0ff28da64bfeaa40a4de2bae03b584a0aab46f9fe2187d5e27be94","first_computed_at":"2026-05-18T04:03:43.129680Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:43.129680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FS6Wg4754QjUHafHH0qYPchuJGY3hIiswEvY27qJaghO+rLrWoLG+rx9OzpageZTmmTzzkIu1xOsypuyo0dmDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:43.130495Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.5225","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:954dff6b6dd7f6ba7ee4cd1f607640bebd9cd47db39e900d74f3e6bdf19ba932","sha256:70f8ea80c08c2df099add0d4b1563faf4b62916282b207ec16f1a2455e316aca"],"state_sha256":"4bcb3befe97bf29f60db63c18028042b9ebbcbf6e38d470d9a1cdb89737cb0b4"}