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Then the dual $G^*$ is a cubic 3-connected planar graph, and $G^*$ is bipartite if and only if $G$ is Eulerian. We prove that if the vertices of $G$ are (improperly) coloured blue and red, such that the blue vertices cover the faces of $G$, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then $G^*$ is Hamiltonian.\n  This result implies the following special case "},"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":"1312.3783","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T11:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"02ea7711bca38c198baa35244f74f70d0eaf119addade57cc60b63e67d3db86e","abstract_canon_sha256":"e6574c1e5cfe8958c338f094351b8f8c8d09d8358e06cfef84dcf7a10094dbdc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:47.659745Z","signature_b64":"kffn/0WWcZGcR1kyc56ZPgYyqDSMlUyHnQFD+anXD4bE7fU3O7SUBfe99vqS6Bel05Mutzxs5A+2YtEAYfC+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab1d19ff083ae8fb8f3289a3571e4f1f067d276875308dd7d56a089ad87defdf","last_reissued_at":"2026-05-18T03:04:47.658987Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:47.658987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Thoughts on Barnette's Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Helmut Alt, Jens M. Schmidt, Michael S. Payne","submitted_at":"2013-12-13T11:53:29Z","abstract_excerpt":"We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic 3-connected planar graph, and $G^*$ is bipartite if and only if $G$ is Eulerian. We prove that if the vertices of $G$ are (improperly) coloured blue and red, such that the blue vertices cover the faces of $G$, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then $G^*$ is Hamiltonian.\n  This result implies the following special case "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3783","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1312.3783","created_at":"2026-05-18T03:04:47.659110+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.3783v1","created_at":"2026-05-18T03:04:47.659110+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3783","created_at":"2026-05-18T03:04:47.659110+00:00"},{"alias_kind":"pith_short_12","alias_value":"VMORT7YIHLUP","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"VMORT7YIHLUPXDZS","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"VMORT7YI","created_at":"2026-05-18T12:28:04.890932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4","json":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4.json","graph_json":"https://pith.science/api/pith-number/VMORT7YIHLUPXDZSRGRVOHSPD4/graph.json","events_json":"https://pith.science/api/pith-number/VMORT7YIHLUPXDZSRGRVOHSPD4/events.json","paper":"https://pith.science/paper/VMORT7YI"},"agent_actions":{"view_html":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4","download_json":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4.json","view_paper":"https://pith.science/paper/VMORT7YI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.3783&json=true","fetch_graph":"https://pith.science/api/pith-number/VMORT7YIHLUPXDZSRGRVOHSPD4/graph.json","fetch_events":"https://pith.science/api/pith-number/VMORT7YIHLUPXDZSRGRVOHSPD4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4/action/storage_attestation","attest_author":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4/action/author_attestation","sign_citation":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4/action/citation_signature","submit_replication":"https://pith.science/pith/VMORT7YIHLUPXDZSRGRVOHSPD4/action/replication_record"}},"created_at":"2026-05-18T03:04:47.659110+00:00","updated_at":"2026-05-18T03:04:47.659110+00:00"}