{"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"}