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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 "},"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":"2606.12271","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T16:11:41Z","cross_cats_sorted":[],"title_canon_sha256":"7afea346844e703808933775d2c688258ca5ccf7aa904545131df52012358782","abstract_canon_sha256":"182f5b57fa18b56505bbba74df7d978486b3f6442458e819a21a542652b648a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:58.947481Z","signature_b64":"Lp+5UgWog+o4YYOtoJDpYyd/ek5hBEUhNlD9bM43tYfF71qjCVYhvMJLVGL337c83kPeZgTnJZJnPTjS4vkvDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04dfa9fbaaa51d9740a0200968cc5e60131c654a71f9cbaf567069c4b496d622","last_reissued_at":"2026-06-11T01:10:58.946685Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:58.946685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\overline{d}(G) \\ge \\Delta-1+\\tfrac{3}{n}$. 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