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Let $|G|$ denote the number of vertices of $G$ and $\\Delta=\\Delta(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $\\Delta$-colorable if $G$ satisfies $(|G|+1)/3 \\leq \\Delta < |G|/2$ and none of its components is a $K_{\\Delta +1}$."},"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":"1408.6046","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-26T08:21:58Z","cross_cats_sorted":[],"title_canon_sha256":"a82beee436a3e56ac5c4a80c714ba641664a1042406b54ceb3ae8410f392195d","abstract_canon_sha256":"c612e8d9ad90be0308956b9ef22e93930f3ea89d42f12d1406c10be339022dec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:15.013226Z","signature_b64":"4PiiHKDXtdfUPOxY4L7YU0bp0SdC9A02CoxD9BDKwnlcpeuTB/SZzWXKKEVjFL0V2hnhlY70vxXjh3CfkjHzAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32c42270dbfcbb3c8f13f1a5fa8f38a798023288da28e3d90ab5b57766d8c90","last_reissued_at":"2026-05-18T02:44:15.012667Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:15.012667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equitable Coloring of Graphs with Intermediate Maximum Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bor-Liang Chen, Ko-Wei Lih, Kuo-Ching Huang","submitted_at":"2014-08-26T08:21:58Z","abstract_excerpt":"If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number of vertices of $G$ and $\\Delta=\\Delta(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $\\Delta$-colorable if $G$ satisfies $(|G|+1)/3 \\leq \\Delta < |G|/2$ and none of its components is a $K_{\\Delta +1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6046","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":"1408.6046","created_at":"2026-05-18T02:44:15.012751+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.6046v1","created_at":"2026-05-18T02:44:15.012751+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6046","created_at":"2026-05-18T02:44:15.012751+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MWEEJYNX7F3","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MWEEJYNX7F3HSHR","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MWEEJYN","created_at":"2026-05-18T12:28:14.216126+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/4MWEEJYNX7F3HSHRH4NF7KHTRJ","json":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ.json","graph_json":"https://pith.science/api/pith-number/4MWEEJYNX7F3HSHRH4NF7KHTRJ/graph.json","events_json":"https://pith.science/api/pith-number/4MWEEJYNX7F3HSHRH4NF7KHTRJ/events.json","paper":"https://pith.science/paper/4MWEEJYN"},"agent_actions":{"view_html":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ","download_json":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ.json","view_paper":"https://pith.science/paper/4MWEEJYN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.6046&json=true","fetch_graph":"https://pith.science/api/pith-number/4MWEEJYNX7F3HSHRH4NF7KHTRJ/graph.json","fetch_events":"https://pith.science/api/pith-number/4MWEEJYNX7F3HSHRH4NF7KHTRJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ/action/storage_attestation","attest_author":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ/action/author_attestation","sign_citation":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ/action/citation_signature","submit_replication":"https://pith.science/pith/4MWEEJYNX7F3HSHRH4NF7KHTRJ/action/replication_record"}},"created_at":"2026-05-18T02:44:15.012751+00:00","updated_at":"2026-05-18T02:44:15.012751+00:00"}