{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:4566PQZ6Y57EFAOMEYM74YXZ3S","short_pith_number":"pith:4566PQZ6","canonical_record":{"source":{"id":"0911.4162","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T05:39:48Z","cross_cats_sorted":[],"title_canon_sha256":"da23104bdf92ba391b06a0b6f65e1c148f4c2a0fe60d23398a713fdb48840f0f","abstract_canon_sha256":"3873e6b76915de5bce227a70149e0588ad61570dc4a0847b8ee4fac91596028a"},"schema_version":"1.0"},"canonical_sha256":"e77de7c33ec77e4281cc2619fe62f9dc99fd35c133e7e1d7af21a015492850b1","source":{"kind":"arxiv","id":"0911.4162","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4162","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4162v1","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4162","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"pith_short_12","alias_value":"4566PQZ6Y57E","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"4566PQZ6Y57EFAOM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"4566PQZ6","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:4566PQZ6Y57EFAOMEYM74YXZ3S","target":"record","payload":{"canonical_record":{"source":{"id":"0911.4162","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T05:39:48Z","cross_cats_sorted":[],"title_canon_sha256":"da23104bdf92ba391b06a0b6f65e1c148f4c2a0fe60d23398a713fdb48840f0f","abstract_canon_sha256":"3873e6b76915de5bce227a70149e0588ad61570dc4a0847b8ee4fac91596028a"},"schema_version":"1.0"},"canonical_sha256":"e77de7c33ec77e4281cc2619fe62f9dc99fd35c133e7e1d7af21a015492850b1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:23.750095Z","signature_b64":"/hjVL8KX1pwkm4zmcIZWynoXVdUJplkD8J8Af7cpwo2zz0sY/TfzWvpq8Ow+CJ3tMRcwQ3hTELhizJwFqLUyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e77de7c33ec77e4281cc2619fe62f9dc99fd35c133e7e1d7af21a015492850b1","last_reissued_at":"2026-05-18T03:55:23.749327Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:23.749327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0911.4162","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":"2026-05-18T03:55:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Q0g20X5UPg9qDnNTSrMja0mDeDNS/eMeS47HMn47sg8FhKfdujs1hkeTm+xav0P7vnHnpSbU1m43qYtyVCaBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T07:33:56.649722Z"},"content_sha256":"9197f584e3b45f2cc23d636a4f3bdf8f546c5cc63021564b574351211ee9d6cd","schema_version":"1.0","event_id":"sha256:9197f584e3b45f2cc23d636a4f3bdf8f546c5cc63021564b574351211ee9d6cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:4566PQZ6Y57EFAOMEYM74YXZ3S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the book thickness of $k$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Vida Dujmovi\\'c","submitted_at":"2009-11-21T05:39:48Z","abstract_excerpt":"Every $k$-tree has book thickness at most $k+1$, and this bound is best possible for all $k\\geq3$. Vandenbussche et al. (2009) proved that every $k$-tree that has a smooth degree-3 tree decomposition with width $k$ has book thickness at most $k$. We prove this result is best possible for $k\\geq 4$, by constructing a $k$-tree with book thickness $k+1$ that has a smooth degree-4 tree decomposition with width $k$. This solves an open problem of Vandenbussche et al. (2009)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4162","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":"2026-05-18T03:55:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rG83R1hcmiKc0idF0NDOY9OqgmQ0q0Yh8C/ZYvxQsav4f4NvMpDiiQBFLT5OMbw4lCum0TcTNkIvamHIBJDLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T07:33:56.650441Z"},"content_sha256":"a48194d7b2339bf9c5616fa38250ad81ecb6a1abc935f4f6942285b90fe1ee30","schema_version":"1.0","event_id":"sha256:a48194d7b2339bf9c5616fa38250ad81ecb6a1abc935f4f6942285b90fe1ee30"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/bundle.json","state_url":"https://pith.science/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/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":"2026-05-26T07:33:56Z","links":{"resolver":"https://pith.science/pith/4566PQZ6Y57EFAOMEYM74YXZ3S","bundle":"https://pith.science/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/bundle.json","state":"https://pith.science/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4566PQZ6Y57EFAOMEYM74YXZ3S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:4566PQZ6Y57EFAOMEYM74YXZ3S","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":"3873e6b76915de5bce227a70149e0588ad61570dc4a0847b8ee4fac91596028a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T05:39:48Z","title_canon_sha256":"da23104bdf92ba391b06a0b6f65e1c148f4c2a0fe60d23398a713fdb48840f0f"},"schema_version":"1.0","source":{"id":"0911.4162","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.4162","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"arxiv_version","alias_value":"0911.4162v1","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.4162","created_at":"2026-05-18T03:55:23Z"},{"alias_kind":"pith_short_12","alias_value":"4566PQZ6Y57E","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"4566PQZ6Y57EFAOM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"4566PQZ6","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a48194d7b2339bf9c5616fa38250ad81ecb6a1abc935f4f6942285b90fe1ee30","target":"graph","created_at":"2026-05-18T03:55:23Z","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":"Every $k$-tree has book thickness at most $k+1$, and this bound is best possible for all $k\\geq3$. Vandenbussche et al. (2009) proved that every $k$-tree that has a smooth degree-3 tree decomposition with width $k$ has book thickness at most $k$. We prove this result is best possible for $k\\geq 4$, by constructing a $k$-tree with book thickness $k+1$ that has a smooth degree-4 tree decomposition with width $k$. This solves an open problem of Vandenbussche et al. (2009)","authors_text":"David R. Wood, Vida Dujmovi\\'c","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T05:39:48Z","title":"On the book thickness of $k$-trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4162","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":"sha256:9197f584e3b45f2cc23d636a4f3bdf8f546c5cc63021564b574351211ee9d6cd","target":"record","created_at":"2026-05-18T03:55:23Z","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":"3873e6b76915de5bce227a70149e0588ad61570dc4a0847b8ee4fac91596028a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-11-21T05:39:48Z","title_canon_sha256":"da23104bdf92ba391b06a0b6f65e1c148f4c2a0fe60d23398a713fdb48840f0f"},"schema_version":"1.0","source":{"id":"0911.4162","kind":"arxiv","version":1}},"canonical_sha256":"e77de7c33ec77e4281cc2619fe62f9dc99fd35c133e7e1d7af21a015492850b1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e77de7c33ec77e4281cc2619fe62f9dc99fd35c133e7e1d7af21a015492850b1","first_computed_at":"2026-05-18T03:55:23.749327Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:23.749327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/hjVL8KX1pwkm4zmcIZWynoXVdUJplkD8J8Af7cpwo2zz0sY/TfzWvpq8Ow+CJ3tMRcwQ3hTELhizJwFqLUyAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:23.750095Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.4162","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9197f584e3b45f2cc23d636a4f3bdf8f546c5cc63021564b574351211ee9d6cd","sha256:a48194d7b2339bf9c5616fa38250ad81ecb6a1abc935f4f6942285b90fe1ee30"],"state_sha256":"820f7225859fda52ca41be83340f12a596239b23e4334fe0d92fa49df51692d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r41VgVcmRRpht07W/m81QYemwkrxthBgMdoEam3i4uhHzXhooNiGLgD7j8LkmiuQHUGELDbFCQdymn8dnZIaCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T07:33:56.653724Z","bundle_sha256":"c4f6f26ada04af065294f5c081bd84d6ffddbab9a21603252f01484b75ac11af"}}