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A ${\\rm b}^{\\ast}$-coloring is a variation in which a b-vertex is adjacent to a b-vertex in every other color class. We employ the ${\\rm b}^{\\ast}$-coloring to prove that any $n$-vertex graph $G$ with independence number at most $t$ satisfies ${\\rm b}(G) \\leq ((t-1)n+t\\chi(G))/(2t-1)$. This bound extends the bounds of Kouider and Zaker (20"},"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":"2606.07461","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T17:13:10Z","cross_cats_sorted":[],"title_canon_sha256":"ef98044a6079cb32664a16412a2c34fc33f2a319434bf0a47eaf410a49de7743","abstract_canon_sha256":"a907c2bd0e13e4344144f4a00b90166e66eb8b67b85c0a6dbb0dd553e3dd02c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:05:28.403972Z","signature_b64":"zx4YS+5q5rue29715Cx952ul2pczzQ8kG4Fv5IxuIOgKntuG09KF7eMuXaoEFE41Vg1J6Kr6wZB8+lXMhxSSDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60beccea29ae6bd9c9b539344f856c526c8a4c9ac4a9f606321363d5fe0f28b0","last_reissued_at":"2026-06-08T01:05:28.403002Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:05:28.403002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved bounds on the b-chromatic number using the independence and chromatic numbers","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Manouchehr Zaker","submitted_at":"2026-06-05T17:13:10Z","abstract_excerpt":"A b-coloring of a graph $G$ is a proper vertex coloring where each color class contains at least one vertex (a b-vertex) adjacent to a vertex in every other color class. The maximum number of colors in such a coloring is the b-chromatic number, ${\\rm b}(G)$. A ${\\rm b}^{\\ast}$-coloring is a variation in which a b-vertex is adjacent to a b-vertex in every other color class. We employ the ${\\rm b}^{\\ast}$-coloring to prove that any $n$-vertex graph $G$ with independence number at most $t$ satisfies ${\\rm b}(G) \\leq ((t-1)n+t\\chi(G))/(2t-1)$. 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