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They showed that $2\\le d_0\\le 4$ and $d_1\\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\\{4,5\\}$-cycles and $d^{\\Delta}\\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\\{4,5,6\\}$-cycles"},"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":"1809.00925","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-04T12:53:32Z","cross_cats_sorted":[],"title_canon_sha256":"effc3c9c79f5de111e1c9dbf237797a759471b470405f80f5cd41babbb69cf30","abstract_canon_sha256":"656a801cfbfbb27c998e5e90c0540761e6c4b97729101ba89e8edce7e54dfe54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:32.134424Z","signature_b64":"YvXocgIeaohMfhv7w4F92BGD1Zk7lhcBkv3rAy1/HNYPhtAqgh0oNxafyehLaqPFN0MD1fayxDBWQaBpHBtcBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5aa58f147ff2ba16c638e0185cd05e6dc97dd77a6971d881eacde19079ab6918","last_reissued_at":"2026-05-18T00:06:32.133900Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:32.133900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Yuxue Yin","submitted_at":"2018-09-04T12:53:32Z","abstract_excerpt":"Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\\{4,5\\}$-cycles and $d^{\\Delta}\\ge d_0$ are $3$-choosable and planar graphs without $\\{4,5,6\\}$-cycles and $d^{\\Delta}\\ge d_1$ are $3$-choosable, where $d^{\\Delta}$ is the smallest distance between triangles. They showed that $2\\le d_0\\le 4$ and $d_1\\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\\{4,5\\}$-cycles and $d^{\\Delta}\\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\\{4,5,6\\}$-cycles"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00925","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":"1809.00925","created_at":"2026-05-18T00:06:32.133990+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.00925v1","created_at":"2026-05-18T00:06:32.133990+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00925","created_at":"2026-05-18T00:06:32.133990+00:00"},{"alias_kind":"pith_short_12","alias_value":"LKSY6FD76K5B","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_16","alias_value":"LKSY6FD76K5BNRRY","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_8","alias_value":"LKSY6FD7","created_at":"2026-05-18T12:32:37.024351+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/LKSY6FD76K5BNRRY4AMFZUC6NX","json":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX.json","graph_json":"https://pith.science/api/pith-number/LKSY6FD76K5BNRRY4AMFZUC6NX/graph.json","events_json":"https://pith.science/api/pith-number/LKSY6FD76K5BNRRY4AMFZUC6NX/events.json","paper":"https://pith.science/paper/LKSY6FD7"},"agent_actions":{"view_html":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX","download_json":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX.json","view_paper":"https://pith.science/paper/LKSY6FD7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.00925&json=true","fetch_graph":"https://pith.science/api/pith-number/LKSY6FD76K5BNRRY4AMFZUC6NX/graph.json","fetch_events":"https://pith.science/api/pith-number/LKSY6FD76K5BNRRY4AMFZUC6NX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX/action/storage_attestation","attest_author":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX/action/author_attestation","sign_citation":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX/action/citation_signature","submit_replication":"https://pith.science/pith/LKSY6FD76K5BNRRY4AMFZUC6NX/action/replication_record"}},"created_at":"2026-05-18T00:06:32.133990+00:00","updated_at":"2026-05-18T00:06:32.133990+00:00"}