{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BB6UUDGAZL4WPPSHYYBI7L2LDV","short_pith_number":"pith:BB6UUDGA","canonical_record":{"source":{"id":"1704.06536","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T13:37:54Z","cross_cats_sorted":[],"title_canon_sha256":"200289d11bd1d9a3d71e382502cb23b7ede3da1927e17694bb07d5dde8977f79","abstract_canon_sha256":"9c74657cd3bae192bf979a6b91ce175becb2f04a300924d2e8ba4c06b7792249"},"schema_version":"1.0"},"canonical_sha256":"087d4a0cc0caf967be47c6028faf4b1d51f9df4f70a7fb67a1e1d3db12b83822","source":{"kind":"arxiv","id":"1704.06536","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06536","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06536v4","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06536","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"pith_short_12","alias_value":"BB6UUDGAZL4W","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BB6UUDGAZL4WPPSH","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BB6UUDGA","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BB6UUDGAZL4WPPSHYYBI7L2LDV","target":"record","payload":{"canonical_record":{"source":{"id":"1704.06536","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T13:37:54Z","cross_cats_sorted":[],"title_canon_sha256":"200289d11bd1d9a3d71e382502cb23b7ede3da1927e17694bb07d5dde8977f79","abstract_canon_sha256":"9c74657cd3bae192bf979a6b91ce175becb2f04a300924d2e8ba4c06b7792249"},"schema_version":"1.0"},"canonical_sha256":"087d4a0cc0caf967be47c6028faf4b1d51f9df4f70a7fb67a1e1d3db12b83822","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:49.904265Z","signature_b64":"BnQc/VVUz038dRPDII1wjuVQMwgBxqYcJQ5fr+QyKZv5nZjge7C/qDhLOQZ2ihmEnnreQErJl8HqzDsPYRCpBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"087d4a0cc0caf967be47c6028faf4b1d51f9df4f70a7fb67a1e1d3db12b83822","last_reissued_at":"2026-05-17T23:40:49.903528Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:49.903528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.06536","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/tO9pnF2lkDxe5ozO0cxA0l5dVHM0CXW3kMV41y8tx+aXxPXad+BZO3WHtdsdi8dWHgBHerFMxpv4vXQdxirAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T04:19:12.584863Z"},"content_sha256":"b7a5d14a64f25aaf25386a364e8ad78123f22645b67af0cd80dfd0448b51ac95","schema_version":"1.0","event_id":"sha256:b7a5d14a64f25aaf25386a364e8ad78123f22645b67af0cd80dfd0448b51ac95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BB6UUDGAZL4WPPSHYYBI7L2LDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improper Colourings inspired by Hadwiger's Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Jan van den Heuvel","submitted_at":"2017-04-21T13:37:54Z","abstract_excerpt":"Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with monochromatic components of order at most $\\lceil{\\frac12(t-2)}\\rceil$. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$-minor-free graph is $(t-1)$-colourable with monochromatic degree at most $t-2$. This is the best known degree bound for such a result. Bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06536","kind":"arxiv","version":4},"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":"2026-05-17T23:40:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rj6mBoG4oT4/XfHgMKXyRuPeCH0nsHtWfkB9yWDN/mhFnn0YOskfH21w4U89RLlS5PPbgpQjMnXxMpBPYPyaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T04:19:12.585204Z"},"content_sha256":"e567a5fd5c461c18b7e39c6975b301cd8e595b96ef4bf570fa8c5d8f9851c9d1","schema_version":"1.0","event_id":"sha256:e567a5fd5c461c18b7e39c6975b301cd8e595b96ef4bf570fa8c5d8f9851c9d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/bundle.json","state_url":"https://pith.science/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/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":"2026-06-12T04:19:12Z","links":{"resolver":"https://pith.science/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV","bundle":"https://pith.science/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/bundle.json","state":"https://pith.science/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BB6UUDGAZL4WPPSHYYBI7L2LDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BB6UUDGAZL4WPPSHYYBI7L2LDV","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":"9c74657cd3bae192bf979a6b91ce175becb2f04a300924d2e8ba4c06b7792249","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T13:37:54Z","title_canon_sha256":"200289d11bd1d9a3d71e382502cb23b7ede3da1927e17694bb07d5dde8977f79"},"schema_version":"1.0","source":{"id":"1704.06536","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06536","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06536v4","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06536","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"pith_short_12","alias_value":"BB6UUDGAZL4W","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BB6UUDGAZL4WPPSH","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BB6UUDGA","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:e567a5fd5c461c18b7e39c6975b301cd8e595b96ef4bf570fa8c5d8f9851c9d1","target":"graph","created_at":"2026-05-17T23:40:49Z","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":"Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with monochromatic components of order at most $\\lceil{\\frac12(t-2)}\\rceil$. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every $K_t$-minor-free graph is $(t-1)$-colourable with monochromatic degree at most $t-2$. This is the best known degree bound for such a result. Bo","authors_text":"David R. Wood, Jan van den Heuvel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T13:37:54Z","title":"Improper Colourings inspired by Hadwiger's Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06536","kind":"arxiv","version":4},"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:b7a5d14a64f25aaf25386a364e8ad78123f22645b67af0cd80dfd0448b51ac95","target":"record","created_at":"2026-05-17T23:40:49Z","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":"9c74657cd3bae192bf979a6b91ce175becb2f04a300924d2e8ba4c06b7792249","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T13:37:54Z","title_canon_sha256":"200289d11bd1d9a3d71e382502cb23b7ede3da1927e17694bb07d5dde8977f79"},"schema_version":"1.0","source":{"id":"1704.06536","kind":"arxiv","version":4}},"canonical_sha256":"087d4a0cc0caf967be47c6028faf4b1d51f9df4f70a7fb67a1e1d3db12b83822","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"087d4a0cc0caf967be47c6028faf4b1d51f9df4f70a7fb67a1e1d3db12b83822","first_computed_at":"2026-05-17T23:40:49.903528Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:49.903528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BnQc/VVUz038dRPDII1wjuVQMwgBxqYcJQ5fr+QyKZv5nZjge7C/qDhLOQZ2ihmEnnreQErJl8HqzDsPYRCpBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:49.904265Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.06536","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7a5d14a64f25aaf25386a364e8ad78123f22645b67af0cd80dfd0448b51ac95","sha256:e567a5fd5c461c18b7e39c6975b301cd8e595b96ef4bf570fa8c5d8f9851c9d1"],"state_sha256":"5cd27104bfcb85ac04f163a96bca188ec29ed15057949b8fbee5f4fb0147c817"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2nZAYZqzAoHyLmOwSSEaSwoFsADO1DxY3R7auLyqaiX8wjxmdqgmeDw2YCvD+7JQK52Tk2qvnMOxAWulh/7DCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T04:19:12.587218Z","bundle_sha256":"9672a6893931c0a2c4a7a529b396221a463d237041ec448ca91d71e52992e968"}}