{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:W72CCI53VF5D72K6YUMYFM5IVE","short_pith_number":"pith:W72CCI53","canonical_record":{"source":{"id":"1807.11617","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-31T00:44:34Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"3f761b3d5a15dd32a3f90ffacfb7b3639219b5b512fae1337c78417f3c071f5b","abstract_canon_sha256":"090eb8641a4a9d55c52c5febe362465b2271697f2385b02242072d2b1192901d"},"schema_version":"1.0"},"canonical_sha256":"b7f42123bba97a3fe95ec51982b3a8a93ebd60b1a4175ea428ab94b89c805bf9","source":{"kind":"arxiv","id":"1807.11617","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.11617","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"1807.11617v1","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.11617","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"W72CCI53VF5D","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"W72CCI53VF5D72K6","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"W72CCI53","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:W72CCI53VF5D72K6YUMYFM5IVE","target":"record","payload":{"canonical_record":{"source":{"id":"1807.11617","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-31T00:44:34Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"3f761b3d5a15dd32a3f90ffacfb7b3639219b5b512fae1337c78417f3c071f5b","abstract_canon_sha256":"090eb8641a4a9d55c52c5febe362465b2271697f2385b02242072d2b1192901d"},"schema_version":"1.0"},"canonical_sha256":"b7f42123bba97a3fe95ec51982b3a8a93ebd60b1a4175ea428ab94b89c805bf9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:21.598048Z","signature_b64":"ssS1Xxm0zqkSl1jzJMkXrwEA8XHNpqzkDmszP6OzfZftvLfUwsWdWHwJnilg7oK32zWg7f9gdgykRcCiQpjNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7f42123bba97a3fe95ec51982b3a8a93ebd60b1a4175ea428ab94b89c805bf9","last_reissued_at":"2026-05-18T00:09:21.597450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:21.597450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.11617","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":"2026-05-18T00:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z/YN2xjF0502X7+0g9EnE776puhkjqlDRrdoxaQ6N2DNFds9gYBNnyX91FkeItpcktPHrhq5Xma6P8dxAFZaBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:05:41.213181Z"},"content_sha256":"663b5b87a93dfed9c036bf699b910eabe2e9d8ed42bcb3d990be2e52870a31e9","schema_version":"1.0","event_id":"sha256:663b5b87a93dfed9c036bf699b910eabe2e9d8ed42bcb3d990be2e52870a31e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:W72CCI53VF5D72K6YUMYFM5IVE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tight Upper Bounds on the Crossing Number in a Minor-Closed Class","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Bojan Mohar, David R. Wood, Ken-ichi Kawarabayashi, Vida Dujmovi\\'c","submitted_at":"2018-07-31T00:44:34Z","abstract_excerpt":"The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\\Delta n)$, where $G$ has $n$ vertices and maximum degree $\\Delta$. This dependence on $n$ and $\\Delta$ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of $O(\\Delta^2 n)$. We also study the convex and rectilinear crossing numbers, and prove an $O(\\Delta n)$ bound for the convex crossing num"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11617","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":"2026-05-18T00:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RPiq6YBY66zw61IlJofVKvgz/6UKmjnEDFYBLo7HU9E58rGsAz/Es+AJBgQP6rt1Qy8gH9H0v/jK7uc11i01CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:05:41.213896Z"},"content_sha256":"dba64a98a0020376516d762ccff21ad05b0f4029e8bab9a9c34670f92130d984","schema_version":"1.0","event_id":"sha256:dba64a98a0020376516d762ccff21ad05b0f4029e8bab9a9c34670f92130d984"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W72CCI53VF5D72K6YUMYFM5IVE/bundle.json","state_url":"https://pith.science/pith/W72CCI53VF5D72K6YUMYFM5IVE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W72CCI53VF5D72K6YUMYFM5IVE/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":"2026-05-25T15:05:41Z","links":{"resolver":"https://pith.science/pith/W72CCI53VF5D72K6YUMYFM5IVE","bundle":"https://pith.science/pith/W72CCI53VF5D72K6YUMYFM5IVE/bundle.json","state":"https://pith.science/pith/W72CCI53VF5D72K6YUMYFM5IVE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W72CCI53VF5D72K6YUMYFM5IVE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:W72CCI53VF5D72K6YUMYFM5IVE","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":"090eb8641a4a9d55c52c5febe362465b2271697f2385b02242072d2b1192901d","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-31T00:44:34Z","title_canon_sha256":"3f761b3d5a15dd32a3f90ffacfb7b3639219b5b512fae1337c78417f3c071f5b"},"schema_version":"1.0","source":{"id":"1807.11617","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.11617","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"1807.11617v1","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.11617","created_at":"2026-05-18T00:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"W72CCI53VF5D","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"W72CCI53VF5D72K6","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"W72CCI53","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:dba64a98a0020376516d762ccff21ad05b0f4029e8bab9a9c34670f92130d984","target":"graph","created_at":"2026-05-18T00:09:21Z","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 crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\\Delta n)$, where $G$ has $n$ vertices and maximum degree $\\Delta$. This dependence on $n$ and $\\Delta$ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of $O(\\Delta^2 n)$. We also study the convex and rectilinear crossing numbers, and prove an $O(\\Delta n)$ bound for the convex crossing num","authors_text":"Bojan Mohar, David R. Wood, Ken-ichi Kawarabayashi, Vida Dujmovi\\'c","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-31T00:44:34Z","title":"Tight Upper Bounds on the Crossing Number in a Minor-Closed Class"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11617","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":"sha256:663b5b87a93dfed9c036bf699b910eabe2e9d8ed42bcb3d990be2e52870a31e9","target":"record","created_at":"2026-05-18T00:09:21Z","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":"090eb8641a4a9d55c52c5febe362465b2271697f2385b02242072d2b1192901d","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-31T00:44:34Z","title_canon_sha256":"3f761b3d5a15dd32a3f90ffacfb7b3639219b5b512fae1337c78417f3c071f5b"},"schema_version":"1.0","source":{"id":"1807.11617","kind":"arxiv","version":1}},"canonical_sha256":"b7f42123bba97a3fe95ec51982b3a8a93ebd60b1a4175ea428ab94b89c805bf9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b7f42123bba97a3fe95ec51982b3a8a93ebd60b1a4175ea428ab94b89c805bf9","first_computed_at":"2026-05-18T00:09:21.597450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:21.597450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ssS1Xxm0zqkSl1jzJMkXrwEA8XHNpqzkDmszP6OzfZftvLfUwsWdWHwJnilg7oK32zWg7f9gdgykRcCiQpjNCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:21.598048Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.11617","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:663b5b87a93dfed9c036bf699b910eabe2e9d8ed42bcb3d990be2e52870a31e9","sha256:dba64a98a0020376516d762ccff21ad05b0f4029e8bab9a9c34670f92130d984"],"state_sha256":"f5bddfcee9104f6a286b70e9a3902e8d7e69a9743c03005faaf2ed23a62f91d1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T06EosFV/Y1EfWy6zlY6JOrJ2I3rItXW5bTk4vNhlQshhahIc1dew2W5tRWtKXst8S89wB6zErA0IXLhflnOCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:05:41.218105Z","bundle_sha256":"a169d7bd18d7b38e9b6e6034e34076b2fa0964dae82daa13dc1bca4311e1f8d2"}}