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To investigate the worst-case performance of the upper bound $B$, we study the graph invariant $$\\beta(G) = \\sup_{\\bx \\ne 0} \\frac{B(G_\\bx)}{\\chi_f(G_\\bx)}.$$\n  \\noindent This invariant is examined for various upper bounds $B$ on the fractional chromatic number. In some important cases, this graph invariant is shown to be related to the size"},"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":"1001.3053","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2010-01-18T13:21:04Z","cross_cats_sorted":["math.CO","math.IT"],"title_canon_sha256":"9ac3c4aac2d4746cf09c8ddc74150df1230ac14d358bf429be355092a694748b","abstract_canon_sha256":"67e5f9ffba6f29606ed66a6b8c277b23a318af3a4469f45510edbbd24e6de6a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:26.629576Z","signature_b64":"05dss0S7Bf81lDCKRgGEHK2ShCc8Vb7zXGuHD3h0biaWaKWTtDK0om+Rl+Y9DvWX5o2nlfweX/61Q34j9mq0BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59ecca255f33ef6fec598df2fe51e4e1b3939cf436517fd932c7289ebb38f785","last_reissued_at":"2026-05-18T04:01:26.629058Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:26.629058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some upper bounds on the fractional chromatic number of weighted graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Ashwin Ganesan","submitted_at":"2010-01-18T13:21:04Z","abstract_excerpt":"Given a weighted graph $G_\\bx$, where $(x(v): v \\in V)$ is a non-negative, real-valued weight assigned to the vertices of G, let $B(G_\\bx)$ be an upper bound on the fractional chromatic number of the weighted graph $G_\\bx$; so $\\chi_f(G_\\bx) \\le B(G_\\bx)$. To investigate the worst-case performance of the upper bound $B$, we study the graph invariant $$\\beta(G) = \\sup_{\\bx \\ne 0} \\frac{B(G_\\bx)}{\\chi_f(G_\\bx)}.$$\n  \\noindent This invariant is examined for various upper bounds $B$ on the fractional chromatic number. 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