{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:F3ZYQXVCSZZTDKBFVWCAH2DN5N","short_pith_number":"pith:F3ZYQXVC","canonical_record":{"source":{"id":"1209.2218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-11T04:22:37Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"14343b2249514ab080dd0b0d86eb159ae5d15bcaea3f919c0c3aafeba3979bd3","abstract_canon_sha256":"52eaeae0ff2597a781df9690fda14121fbdef61af76d6b3c9d1f3e34112e7062"},"schema_version":"1.0"},"canonical_sha256":"2ef3885ea2967331a825ad8403e86deb608806298403b272f59dcec87070f5e7","source":{"kind":"arxiv","id":"1209.2218","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2218","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2218v1","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2218","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"pith_short_12","alias_value":"F3ZYQXVCSZZT","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"F3ZYQXVCSZZTDKBF","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"F3ZYQXVC","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:F3ZYQXVCSZZTDKBFVWCAH2DN5N","target":"record","payload":{"canonical_record":{"source":{"id":"1209.2218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-11T04:22:37Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"14343b2249514ab080dd0b0d86eb159ae5d15bcaea3f919c0c3aafeba3979bd3","abstract_canon_sha256":"52eaeae0ff2597a781df9690fda14121fbdef61af76d6b3c9d1f3e34112e7062"},"schema_version":"1.0"},"canonical_sha256":"2ef3885ea2967331a825ad8403e86deb608806298403b272f59dcec87070f5e7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:52.531766Z","signature_b64":"pz/0QKSuo6VKaA6qeaPCn/FQCrB2ufmZ5dWVGg1eQTG+xk1DFUU+V8bVqKG4N4WVj0V2BmsUgPJ3dyALWo7ICQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ef3885ea2967331a825ad8403e86deb608806298403b272f59dcec87070f5e7","last_reissued_at":"2026-05-18T03:45:52.531277Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:52.531277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.2218","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:45:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jHBkme+3U8+KwfW1yfTPLWBCSMzZTNfIVkkip8aYs2P/V2FR8eFCt4ZrGRfU80BLR0oQmGxoSCCsgRbmJZPlCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:00:14.532019Z"},"content_sha256":"a0a90f819c217d264174e3170212f6324e56b7f6418afe81ac60a8a8675b8a82","schema_version":"1.0","event_id":"sha256:a0a90f819c217d264174e3170212f6324e56b7f6418afe81ac60a8a8675b8a82"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:F3ZYQXVCSZZTDKBFVWCAH2DN5N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Product Dimension of Forests and Bounded Treewidth Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Deepak Rajendraprasad, L. Sunil Chandran, Rogers Mathew, Roohani Sharma","submitted_at":"2012-09-11T04:22:37Z","abstract_excerpt":"The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n verti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2218","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:45:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PmqQfxOElC/z6qlxckTze7yznyU0DDgdiHBIUrjuxtKJ9dMYB5uy4B2nSsZ0rKqlyqM4GgEwSqVeB5U8xhtsAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:00:14.532385Z"},"content_sha256":"5eb234b631b5015126495d3151c2d87e609a131f9543336ca1928114e7a8e29a","schema_version":"1.0","event_id":"sha256:5eb234b631b5015126495d3151c2d87e609a131f9543336ca1928114e7a8e29a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/bundle.json","state_url":"https://pith.science/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T17:00:14Z","links":{"resolver":"https://pith.science/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N","bundle":"https://pith.science/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/bundle.json","state":"https://pith.science/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F3ZYQXVCSZZTDKBFVWCAH2DN5N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:F3ZYQXVCSZZTDKBFVWCAH2DN5N","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":"52eaeae0ff2597a781df9690fda14121fbdef61af76d6b3c9d1f3e34112e7062","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-11T04:22:37Z","title_canon_sha256":"14343b2249514ab080dd0b0d86eb159ae5d15bcaea3f919c0c3aafeba3979bd3"},"schema_version":"1.0","source":{"id":"1209.2218","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2218","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2218v1","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2218","created_at":"2026-05-18T03:45:52Z"},{"alias_kind":"pith_short_12","alias_value":"F3ZYQXVCSZZT","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"F3ZYQXVCSZZTDKBF","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"F3ZYQXVC","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:5eb234b631b5015126495d3151c2d87e609a131f9543336ca1928114e7a8e29a","target":"graph","created_at":"2026-05-18T03:45:52Z","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":"The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n verti","authors_text":"Deepak Rajendraprasad, L. Sunil Chandran, Rogers Mathew, Roohani Sharma","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-11T04:22:37Z","title":"Product Dimension of Forests and Bounded Treewidth Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2218","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:a0a90f819c217d264174e3170212f6324e56b7f6418afe81ac60a8a8675b8a82","target":"record","created_at":"2026-05-18T03:45:52Z","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":"52eaeae0ff2597a781df9690fda14121fbdef61af76d6b3c9d1f3e34112e7062","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-11T04:22:37Z","title_canon_sha256":"14343b2249514ab080dd0b0d86eb159ae5d15bcaea3f919c0c3aafeba3979bd3"},"schema_version":"1.0","source":{"id":"1209.2218","kind":"arxiv","version":1}},"canonical_sha256":"2ef3885ea2967331a825ad8403e86deb608806298403b272f59dcec87070f5e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ef3885ea2967331a825ad8403e86deb608806298403b272f59dcec87070f5e7","first_computed_at":"2026-05-18T03:45:52.531277Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:52.531277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pz/0QKSuo6VKaA6qeaPCn/FQCrB2ufmZ5dWVGg1eQTG+xk1DFUU+V8bVqKG4N4WVj0V2BmsUgPJ3dyALWo7ICQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:52.531766Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.2218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0a90f819c217d264174e3170212f6324e56b7f6418afe81ac60a8a8675b8a82","sha256:5eb234b631b5015126495d3151c2d87e609a131f9543336ca1928114e7a8e29a"],"state_sha256":"56a3aba9a34fe5f699789746d6cabbda1048f0ee9f5985ed0ff976b07dff8cdd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IiTNZPg0MoAyGCb0jJ3DQ8I2CdXAjCyEJ3gJFSCSsEtozG1Pe8HSkRvojjvRd/mcCpahRzl260i/HCT/khLnBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T17:00:14.534437Z","bundle_sha256":"88d405f8c5e6d72d3921149b93855011d66bd2e591c2a7083deb3a73a53346be"}}