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A simple graph $G$ is called {\\it edge-$\\Delta$-critical} if $\\chi^\\prime(G)=\\Delta+1$ and $\\chi^\\prime(H)\\le\\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$\\Delta$-critical, then $\\bar{d}\\geq \\Delta-1+ \\frac{3}{n}$. We show that $$ \\begin{displaystyle} \\avd \\ge \\begin{cases}\n  0.69241\\D-0.15658 \\quad\\,\\: \\mbox{ if } \\Delta\\geq 66,\n  0.69392\\D-0.20642\\quad\\;\\,\\mbox{ if } \\Delta=65, \\mbox{ and }\n  0."},"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":"1708.01279","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-03T18:48:16Z","cross_cats_sorted":[],"title_canon_sha256":"55030c9548b34f859ee79730f7aab5a287cf407216e4f61995600fa9be9fd4f8","abstract_canon_sha256":"0ea8f0ac230e2fd7d49fe92bbedf3ec5b820cc8945bb7548fb27e4b0e647e973"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:39.460990Z","signature_b64":"29MHQCq/X83Rw0BBoAI8C+XU0+KXIp4QciQt//kYJ3sSh8JX5qyG+abdzJ4D3SazLmk+uE8sMOv1ZC5qHDj/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aebf29eb1ee5eba088af66837c7a4583cdc6b04c752091bebce07dd930d656c1","last_reissued_at":"2026-05-18T00:38:39.460549Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:39.460549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Average degrees of edge-chromatic critical graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fuliang Lu, Guantao Chen, Huiqing Liu, Suyun Jiang, Yan Cao","submitted_at":"2017-08-03T18:48:16Z","abstract_excerpt":"Given a graph $G$, denote by $\\Delta$, $\\bar{d}$ and $\\chi^\\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\\it edge-$\\Delta$-critical} if $\\chi^\\prime(G)=\\Delta+1$ and $\\chi^\\prime(H)\\le\\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$\\Delta$-critical, then $\\bar{d}\\geq \\Delta-1+ \\frac{3}{n}$. We show that $$ \\begin{displaystyle} \\avd \\ge \\begin{cases}\n  0.69241\\D-0.15658 \\quad\\,\\: \\mbox{ if } \\Delta\\geq 66,\n  0.69392\\D-0.20642\\quad\\;\\,\\mbox{ if } \\Delta=65, \\mbox{ and }\n  0."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01279","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":"1708.01279","created_at":"2026-05-18T00:38:39.460619+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.01279v1","created_at":"2026-05-18T00:38:39.460619+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01279","created_at":"2026-05-18T00:38:39.460619+00:00"},{"alias_kind":"pith_short_12","alias_value":"V27ST2Y64XV2","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"V27ST2Y64XV2BCFP","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"V27ST2Y6","created_at":"2026-05-18T12:31:49.984773+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/V27ST2Y64XV2BCFPM2BXY6SFQP","json":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP.json","graph_json":"https://pith.science/api/pith-number/V27ST2Y64XV2BCFPM2BXY6SFQP/graph.json","events_json":"https://pith.science/api/pith-number/V27ST2Y64XV2BCFPM2BXY6SFQP/events.json","paper":"https://pith.science/paper/V27ST2Y6"},"agent_actions":{"view_html":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP","download_json":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP.json","view_paper":"https://pith.science/paper/V27ST2Y6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.01279&json=true","fetch_graph":"https://pith.science/api/pith-number/V27ST2Y64XV2BCFPM2BXY6SFQP/graph.json","fetch_events":"https://pith.science/api/pith-number/V27ST2Y64XV2BCFPM2BXY6SFQP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP/action/storage_attestation","attest_author":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP/action/author_attestation","sign_citation":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP/action/citation_signature","submit_replication":"https://pith.science/pith/V27ST2Y64XV2BCFPM2BXY6SFQP/action/replication_record"}},"created_at":"2026-05-18T00:38:39.460619+00:00","updated_at":"2026-05-18T00:38:39.460619+00:00"}