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Borodin and Raspaud conjecture that every planar graph without intersecting triangles and $5$-cycles is $3$-colorable. We prove in this paper that every planar graph without intersecting triangles and $5$-cycles is (2,0,0)-colorable."},"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":"1407.5138","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-19T00:59:07Z","cross_cats_sorted":[],"title_canon_sha256":"731a5218d33b94027a3049c401e2b0d63aa3122524a4f954322dfa8a914fb8a2","abstract_canon_sha256":"9d1233f613373483d0654c4fb071a7910215bb94fc6e5cbe0910b6d1532d2664"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:36.134004Z","signature_b64":"lxbK7/MPQjERFdHMxssxyRKQGJkJ92rJ71F9v8cs2CX/+qACxhh4Yq8si1L7+I3KDQZh/+ysHtmkkPzoKz6MBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"250b025e3017ddd9bebe4d5a07784a747e6e62bb723b83db6ec0e85e75511989","last_reissued_at":"2026-05-18T02:19:36.133503Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:36.133503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A relaxation of the Bordeaux Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Runrun Liu, Xiangwen Li","submitted_at":"2014-07-19T00:59:07Z","abstract_excerpt":"A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\\varphi:V(G)\\mapsto\\{1,2,...,k\\}$ such that for every $i,1 \\leq i \\leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph without intersecting triangles and $5$-cycles is $3$-colorable. 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