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Appearing in different forms, Gupta\\,(1967), Goldberg\\,(1973), Andersen\\,(1977), and Seymour\\,(1979) made the following conjecture: Every multigraph $G$ satisfies $\\chi'(G) \\le \\max\\{ \\Delta(G) + 1, \\Gamma(G) \\}$, where $\\Gamma(G) = \\max_{H \\subseteq G, |V(H)|\\geq 2} \\left\\lceil \\frac{ |E(H)| }{ \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor} \\right\\rceil$ is the density of $G$. In this paper, we present a polynomial-time algorithm for coloring any multigraph with $\\max\\{ \\Delta(G) + 1, \\Gamma(G) \\}$ co"},"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":"2308.15588","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-08-29T19:32:21Z","cross_cats_sorted":[],"title_canon_sha256":"182f1e26f9cb17d62c7d76df45a9c3416d4389724a51f83dc4157d61a2fb724b","abstract_canon_sha256":"550b0abfedc6c220cebdfa2782dc4bb3b0bcbda68b7e7799531180266a752a00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:03:28.044478Z","signature_b64":"bnFGxo5udwlVTQuEivC14CPe8LMQXx1FXQMgGN27SgL3zK88+vpNmOrahvP8kd025r0+3j6X44T2eOtL3l5dAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa3a0064dc99832d6219c78dd1a23f554447d4cf44119df42f13aba9c6f6c8ba","last_reissued_at":"2026-06-02T01:03:28.044008Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:03:28.044008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Edge Coloring of Multigraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guangming Jing","submitted_at":"2023-08-29T19:32:21Z","abstract_excerpt":"Let $\\Delta(G)$ and $\\chi'(G)$ be the maximum degree and chromatic index of a graph $G$, respectively. Appearing in different forms, Gupta\\,(1967), Goldberg\\,(1973), Andersen\\,(1977), and Seymour\\,(1979) made the following conjecture: Every multigraph $G$ satisfies $\\chi'(G) \\le \\max\\{ \\Delta(G) + 1, \\Gamma(G) \\}$, where $\\Gamma(G) = \\max_{H \\subseteq G, |V(H)|\\geq 2} \\left\\lceil \\frac{ |E(H)| }{ \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor} \\right\\rceil$ is the density of $G$. In this paper, we present a polynomial-time algorithm for coloring any multigraph with $\\max\\{ \\Delta(G) + 1, \\Gamma(G) \\}$ co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.15588","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.15588/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2308.15588","created_at":"2026-06-02T01:03:28.044079+00:00"},{"alias_kind":"arxiv_version","alias_value":"2308.15588v6","created_at":"2026-06-02T01:03:28.044079+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.15588","created_at":"2026-06-02T01:03:28.044079+00:00"},{"alias_kind":"pith_short_12","alias_value":"VI5AAZG4TGBS","created_at":"2026-06-02T01:03:28.044079+00:00"},{"alias_kind":"pith_short_16","alias_value":"VI5AAZG4TGBS2YQZ","created_at":"2026-06-02T01:03:28.044079+00:00"},{"alias_kind":"pith_short_8","alias_value":"VI5AAZG4","created_at":"2026-06-02T01:03:28.044079+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/VI5AAZG4TGBS2YQZY6G5DIR7KV","json":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV.json","graph_json":"https://pith.science/api/pith-number/VI5AAZG4TGBS2YQZY6G5DIR7KV/graph.json","events_json":"https://pith.science/api/pith-number/VI5AAZG4TGBS2YQZY6G5DIR7KV/events.json","paper":"https://pith.science/paper/VI5AAZG4"},"agent_actions":{"view_html":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV","download_json":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV.json","view_paper":"https://pith.science/paper/VI5AAZG4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2308.15588&json=true","fetch_graph":"https://pith.science/api/pith-number/VI5AAZG4TGBS2YQZY6G5DIR7KV/graph.json","fetch_events":"https://pith.science/api/pith-number/VI5AAZG4TGBS2YQZY6G5DIR7KV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV/action/storage_attestation","attest_author":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV/action/author_attestation","sign_citation":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV/action/citation_signature","submit_replication":"https://pith.science/pith/VI5AAZG4TGBS2YQZY6G5DIR7KV/action/replication_record"}},"created_at":"2026-06-02T01:03:28.044079+00:00","updated_at":"2026-06-02T01:03:28.044079+00:00"}