{"paper":{"title":"Average degrees of edge-$\\Delta$-critical multigraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guantao Chen, Shengze Wang, Yimo Su, Yuying Ma","submitted_at":"2026-06-10T16:11:41Z","abstract_excerpt":"Let $G$ be a loopless multigraph with maximum degree $\\Delta(G)$, average degree $\\overline{d}(G)$, density $\\Gamma(G)$, and chromatic index $\\chi'(G)$. A multigraph $G$ is called edge-$\\Delta$-critical if $\\Delta(G)=\\Delta$, $\\chi'(G)=\\Delta(G)+1$ and $\\chi'(H) \\le \\Delta(G)$ for every proper subgraph $H\\subset G$. Vizing conjectured that if $G$ is an edge-$\\Delta$-critical simple graph on $n$ vertices, then $\\overline{d}(G) \\ge \\Delta-1+\\tfrac{3}{n}$. Motivated by this, we conjecture that every edge-$\\Delta$-critical multigraph $G$ satisfies $\\overline{d}(G) \\ge \\tfrac{2\\Delta+2}{3}$, which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12271","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12271/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"}