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A function $\\varphi:E(G)\\rightarrow \\{1,...,t\\}$ is called a proper edge $t$-coloring of a graph $G$, if adjacent edges are colored differently and each of $t$ colors is used. The least value of $t$ for which there exists a proper edge $t$-coloring of a graph $G$ is denoted by $\\chi'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $\\chi'(G)\\leq t\\leq |E(G)|$, we denote by $\\alpha(G,t)$ the set of all proper edge $t$-colorings of $G$."},"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":"1308.3322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-08-15T07:07:50Z","cross_cats_sorted":[],"title_canon_sha256":"e90157339caa022ef17925f64498b3fd2dae3e141daa386372ed0fbca9ef4d99","abstract_canon_sha256":"99a7603aef8de3e0db7067833bac76e78a5ff4d435ceae86c66a87a50b4d2ad4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:55.044990Z","signature_b64":"1qZe1Lr+N4Vm65DXmKTivEZVj1BoXh2xH5ApyBZ4dgp/QF5f9tKf7fL4atJKt0qW1nZzek2WYchKVQzzeZCqDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbe0d1cf288dd881bb609176873a45a2e3068f7708528a7b391d576e59e27b70","last_reissued_at":"2026-05-18T03:15:55.044208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:55.044208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some remarks on relations between the $\\mu$-parameters of regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"N.N. 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For any graph $G$, and for any integer $t$ satisfying the inequality $\\chi'(G)\\leq t\\leq |E(G)|$, we denote by $\\alpha(G,t)$ the set of all proper edge $t$-colorings of $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3322","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":"1308.3322","created_at":"2026-05-18T03:15:55.044342+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.3322v1","created_at":"2026-05-18T03:15:55.044342+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3322","created_at":"2026-05-18T03:15:55.044342+00:00"},{"alias_kind":"pith_short_12","alias_value":"7PQNDTZIRXMI","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"7PQNDTZIRXMIDO3A","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"7PQNDTZI","created_at":"2026-05-18T12:27:36.564083+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/7PQNDTZIRXMIDO3ASF3IOOSFUL","json":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL.json","graph_json":"https://pith.science/api/pith-number/7PQNDTZIRXMIDO3ASF3IOOSFUL/graph.json","events_json":"https://pith.science/api/pith-number/7PQNDTZIRXMIDO3ASF3IOOSFUL/events.json","paper":"https://pith.science/paper/7PQNDTZI"},"agent_actions":{"view_html":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL","download_json":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL.json","view_paper":"https://pith.science/paper/7PQNDTZI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.3322&json=true","fetch_graph":"https://pith.science/api/pith-number/7PQNDTZIRXMIDO3ASF3IOOSFUL/graph.json","fetch_events":"https://pith.science/api/pith-number/7PQNDTZIRXMIDO3ASF3IOOSFUL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL/action/storage_attestation","attest_author":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL/action/author_attestation","sign_citation":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL/action/citation_signature","submit_replication":"https://pith.science/pith/7PQNDTZIRXMIDO3ASF3IOOSFUL/action/replication_record"}},"created_at":"2026-05-18T03:15:55.044342+00:00","updated_at":"2026-05-18T03:15:55.044342+00:00"}