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The maximum average degree, $mad(G)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a, b$, we show that if $mad(G)< \\frac{4}{3}a + b$, then $G$ is $(1_1, 1_2, \\ldots, 1_a, 0_1, \\ldots, 0_b)$-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":"1806.07021","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-19T02:39:00Z","cross_cats_sorted":[],"title_canon_sha256":"d09fc39643f0bb20e623acfa2708ad390fcffaacd210fb51995a0a52833b7b50","abstract_canon_sha256":"a706a745d2b4f0cd27c8b4bcd6dc0ede9b0d2e58f5fe9e91b070dc11b028f6a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:57.699149Z","signature_b64":"Uw27ew5JtCTlHLVsyElYWiPpquBHOodM90iF8FPvx7p717zm13Eqqdfj5jnHzg89KK4YkqEQjG9dN3s0i1HvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4ad2851d675b50b3dcb1002b75169322994e94b63b761af4b64942f6405d6e0","last_reissued_at":"2026-05-18T00:12:57.698577Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:57.698577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximum average degree and relaxed coloring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gexin Yu, Michael Kopreski","submitted_at":"2018-06-19T02:39:00Z","abstract_excerpt":"We say a graph is $(d, d, \\ldots, d, 0, \\ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\\ldots,O_b$ and $a$ sets $D_1, D_2,\\ldots, D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad(G)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a, b$, we show that if $mad(G)< \\frac{4}{3}a + b$, then $G$ is $(1_1, 1_2, \\ldots, 1_a, 0_1, \\ldots, 0_b)$-colorable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07021","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":"1806.07021","created_at":"2026-05-18T00:12:57.698663+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.07021v1","created_at":"2026-05-18T00:12:57.698663+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.07021","created_at":"2026-05-18T00:12:57.698663+00:00"},{"alias_kind":"pith_short_12","alias_value":"YSWSQUOWOW2Q","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"YSWSQUOWOW2QWPOL","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"YSWSQUOW","created_at":"2026-05-18T12:33:04.347982+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/YSWSQUOWOW2QWPOLCABLOULJGI","json":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI.json","graph_json":"https://pith.science/api/pith-number/YSWSQUOWOW2QWPOLCABLOULJGI/graph.json","events_json":"https://pith.science/api/pith-number/YSWSQUOWOW2QWPOLCABLOULJGI/events.json","paper":"https://pith.science/paper/YSWSQUOW"},"agent_actions":{"view_html":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI","download_json":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI.json","view_paper":"https://pith.science/paper/YSWSQUOW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.07021&json=true","fetch_graph":"https://pith.science/api/pith-number/YSWSQUOWOW2QWPOLCABLOULJGI/graph.json","fetch_events":"https://pith.science/api/pith-number/YSWSQUOWOW2QWPOLCABLOULJGI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI/action/storage_attestation","attest_author":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI/action/author_attestation","sign_citation":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI/action/citation_signature","submit_replication":"https://pith.science/pith/YSWSQUOWOW2QWPOLCABLOULJGI/action/replication_record"}},"created_at":"2026-05-18T00:12:57.698663+00:00","updated_at":"2026-05-18T00:12:57.698663+00:00"}