{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:G6WCQT3OXCUCZ6YET6GXQBOVMM","short_pith_number":"pith:G6WCQT3O","canonical_record":{"source":{"id":"1607.06718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-22T15:57:40Z","cross_cats_sorted":[],"title_canon_sha256":"448580aaf8a6205472e644f8f3817b8bc1d6aba8c1fd59a773b9073fcc0b727f","abstract_canon_sha256":"cdeff879e97b95a1bc1f97cef771e79f16d2746ffc71b1e11bc538774231cf10"},"schema_version":"1.0"},"canonical_sha256":"37ac284f6eb8a82cfb049f8d7805d5632aae4f11fcb520877231ff5b5b813a70","source":{"kind":"arxiv","id":"1607.06718","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06718","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06718v2","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06718","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"pith_short_12","alias_value":"G6WCQT3OXCUC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G6WCQT3OXCUCZ6YE","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G6WCQT3O","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:G6WCQT3OXCUCZ6YET6GXQBOVMM","target":"record","payload":{"canonical_record":{"source":{"id":"1607.06718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-22T15:57:40Z","cross_cats_sorted":[],"title_canon_sha256":"448580aaf8a6205472e644f8f3817b8bc1d6aba8c1fd59a773b9073fcc0b727f","abstract_canon_sha256":"cdeff879e97b95a1bc1f97cef771e79f16d2746ffc71b1e11bc538774231cf10"},"schema_version":"1.0"},"canonical_sha256":"37ac284f6eb8a82cfb049f8d7805d5632aae4f11fcb520877231ff5b5b813a70","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:37.204702Z","signature_b64":"i36ezsC6b0jfej6K27z3eEKH5w2skSILkIj16SwJb+5j28wwlliJ1B+H4Z0Mbi/Q9UzI9PDehsPNH/Map87pAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37ac284f6eb8a82cfb049f8d7805d5632aae4f11fcb520877231ff5b5b813a70","last_reissued_at":"2026-05-18T00:48:37.204167Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:37.204167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.06718","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:48:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3FQcXY3/4Bo8QhjwygF0oFI7kWsFWE/GNuaSpCzVY+KBrCKXgwrGWRtUjB6sm1ybXqVUyzzAkPDoavnzEMbyDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:00:31.669053Z"},"content_sha256":"cee60f6d42031ce59c6da51e04a318d6f377466824cadb566f274df2c0612969","schema_version":"1.0","event_id":"sha256:cee60f6d42031ce59c6da51e04a318d6f377466824cadb566f274df2c0612969"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:G6WCQT3OXCUCZ6YET6GXQBOVMM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hadwiger's conjecture for graphs with forbidden holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brian Thomas, Zi-Xia Song","submitted_at":"2016-07-22T15:57:40Z","abstract_excerpt":"Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\\ge \\chi(G)$, where $\\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\\alpha(G)\\ge3$ has no hole of length between $4$ and $2\\alpha(G)-1$, then $h(G)\\ge\\chi(G)$. We also prove that if a graph $G$ w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06718","kind":"arxiv","version":2},"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":"2026-05-18T00:48:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XFkGAMYORy/tW6oaYoUP5AWIpM7exf1sg06+bAt07juIUrmVnBe/n2vVTHst5hNwutPKRHNJix+9OjC/YzmPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:00:31.669415Z"},"content_sha256":"869e29ff05a79c6c0c4509febf3c1d1090a98489d1b42bc8bc56d77ded8d2972","schema_version":"1.0","event_id":"sha256:869e29ff05a79c6c0c4509febf3c1d1090a98489d1b42bc8bc56d77ded8d2972"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/bundle.json","state_url":"https://pith.science/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/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":"2026-06-03T04:00:31Z","links":{"resolver":"https://pith.science/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM","bundle":"https://pith.science/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/bundle.json","state":"https://pith.science/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G6WCQT3OXCUCZ6YET6GXQBOVMM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:G6WCQT3OXCUCZ6YET6GXQBOVMM","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":"cdeff879e97b95a1bc1f97cef771e79f16d2746ffc71b1e11bc538774231cf10","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-22T15:57:40Z","title_canon_sha256":"448580aaf8a6205472e644f8f3817b8bc1d6aba8c1fd59a773b9073fcc0b727f"},"schema_version":"1.0","source":{"id":"1607.06718","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06718","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06718v2","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06718","created_at":"2026-05-18T00:48:37Z"},{"alias_kind":"pith_short_12","alias_value":"G6WCQT3OXCUC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G6WCQT3OXCUCZ6YE","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G6WCQT3O","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:869e29ff05a79c6c0c4509febf3c1d1090a98489d1b42bc8bc56d77ded8d2972","target":"graph","created_at":"2026-05-18T00:48:37Z","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":"Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\\ge \\chi(G)$, where $\\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\\alpha(G)\\ge3$ has no hole of length between $4$ and $2\\alpha(G)-1$, then $h(G)\\ge\\chi(G)$. We also prove that if a graph $G$ w","authors_text":"Brian Thomas, Zi-Xia Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-22T15:57:40Z","title":"Hadwiger's conjecture for graphs with forbidden holes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06718","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:cee60f6d42031ce59c6da51e04a318d6f377466824cadb566f274df2c0612969","target":"record","created_at":"2026-05-18T00:48:37Z","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":"cdeff879e97b95a1bc1f97cef771e79f16d2746ffc71b1e11bc538774231cf10","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-22T15:57:40Z","title_canon_sha256":"448580aaf8a6205472e644f8f3817b8bc1d6aba8c1fd59a773b9073fcc0b727f"},"schema_version":"1.0","source":{"id":"1607.06718","kind":"arxiv","version":2}},"canonical_sha256":"37ac284f6eb8a82cfb049f8d7805d5632aae4f11fcb520877231ff5b5b813a70","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37ac284f6eb8a82cfb049f8d7805d5632aae4f11fcb520877231ff5b5b813a70","first_computed_at":"2026-05-18T00:48:37.204167Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:37.204167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i36ezsC6b0jfej6K27z3eEKH5w2skSILkIj16SwJb+5j28wwlliJ1B+H4Z0Mbi/Q9UzI9PDehsPNH/Map87pAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:37.204702Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06718","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cee60f6d42031ce59c6da51e04a318d6f377466824cadb566f274df2c0612969","sha256:869e29ff05a79c6c0c4509febf3c1d1090a98489d1b42bc8bc56d77ded8d2972"],"state_sha256":"bd98eebbfd8d9c965ae2ca356b5455d5e5be0124817fafa6a4dd41295bf12f8b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4qHK6Gsz9yOaloZymKVJsLx4fmZOSjJvZi0IlUGC5bH5xcm4tBi3zEvXrOUu9YAllSGg9epAbw0n+h7qb2i4Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:00:31.671491Z","bundle_sha256":"0ba15e06a5f8b24b22dbc03a70f8be42de8989c787fa661730100e68699c3e3a"}}