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The graph $G$ is called $\\Delta$-critical if $G$ is connected, $\\chi'(G )=\\Delta(G)+1$ and for any $e\\in E(G)$, $\\chi'(G-e)=\\Delta(G)$. Let $G$ be an $n$-vertex $\\Delta$-critical graph. Vizing conjectured that $\\alpha(G)$, the independence number of $G$, is at most $\\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $\\alpha(G)<\\frac{3n}{5}$. We show that for any given $\\varep"},"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":"1805.05996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T19:00:27Z","cross_cats_sorted":[],"title_canon_sha256":"efcb72b85c70b4cc3b96bd5548a58ee564b10976d31f9104c4015bbd6c7418db","abstract_canon_sha256":"25d54b90d04e40ed7f79490217a3d56f79b7833b838526b527a124998b59fc00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:49.205930Z","signature_b64":"a39h7jivXpoy9y3UynrfeJdm+20rqbD0HCIbQZ8XwJ23wcaTllGuP8SC2+wYGJ9PUnuGfhcmxPSluUK1WjZeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9208c6f1c80a5b60b35f79884f0215118395d49e0b04577de2e1ab4fe84a5e05","last_reissued_at":"2026-05-18T00:15:49.205384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:49.205384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Independence number of edge-chromatic critical graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guangming Jing, Guantao Chen, Songling Shan, Yan Cao","submitted_at":"2018-05-15T19:00:27Z","abstract_excerpt":"Let $G$ be a simple graph with maximum degree $\\Delta(G)$ and chromatic index $\\chi'(G)$. A classic result of Vizing indicates that either $\\chi'(G )=\\Delta(G)$ or $\\chi'(G )=\\Delta(G)+1$. The graph $G$ is called $\\Delta$-critical if $G$ is connected, $\\chi'(G )=\\Delta(G)+1$ and for any $e\\in E(G)$, $\\chi'(G-e)=\\Delta(G)$. Let $G$ be an $n$-vertex $\\Delta$-critical graph. Vizing conjectured that $\\alpha(G)$, the independence number of $G$, is at most $\\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $\\alpha(G)<\\frac{3n}{5}$. 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