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Let $\\mathcal{F}$ denote the family of plane graphs with neither adjacent 3-cycles nor $5$-cycle. Borodin and Raspaud (2003) conjectured that each graph in $\\mathcal{F}$ is $(0,0,0)$-colorable. In this paper, we prove that each graph in $\\mathcal{F}$ is $(1, 1, 0)$-colorable, which improves the results by "},"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":"1508.07890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-31T16:14:00Z","cross_cats_sorted":[],"title_canon_sha256":"df8a54b212706006ac291875b416ad1a47a95542d76508265e2929eca26cb8fb","abstract_canon_sha256":"08cf07d011f25cd3e90c390766f983ed1793cc614edf00e2a2f3fcfc08a1442e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:31.245343Z","signature_b64":"6MiEWe6o6cDJ3WMfgra4h/yCu7ObKvS4XwCS2DNT+b1Qf568UBQqnASSBXGf/Ixd6fWmqFGazMv9eZryOcLRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81e6c977e36973c4e35db3b0a75c89515c47bbd1f870e28384fe62e079ff95ae","last_reissued_at":"2026-05-18T01:34:31.244955Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:31.244955Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A relaxation of the strong Bordeaux Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Xiangwen Li, Ziwen Huang","submitted_at":"2015-08-31T16:14:00Z","abstract_excerpt":"Let $c_1, c_2, \\cdots, c_k$ be $k$ non-negative integers. 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