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This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that the smallest regular triangle-free 5-chromatic graphs have 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-ch"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.07581","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-24T14:43:58Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"4769a4f680c3048f0ae9c2a767cf4675136e599cb8c4960fce4a73e1f9976617","abstract_canon_sha256":"d0d5309376fddee14f2647ad9e507412041ee3b7d0831b84f5e5ea1974ddbb65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:09.974513Z","signature_b64":"345soScIRxfhCMBQb0ur0DdL2avqUzt6l/0YX8cm5MolZyrlPzT+o6cMOGB5uej1le/Wt+3zj/dFTHPRKQxQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"488abf4ae24d326f14817091da42c0ab5e061df5ea297b04b2e8e6c85a9f9193","last_reissued_at":"2026-05-18T00:09:09.973802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:09.973802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On minimal triangle-free 6-chromatic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur","submitted_at":"2017-07-24T14:43:58Z","abstract_excerpt":"A graph with chromatic number $k$ is called $k$-chromatic. 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