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The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow edge-connected. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\\leq 6\\frac{n-2}{\\sigma_2+2}+7$. Moreover, the bound is seen to be tight up to add"},"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":"1101.3119","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-17T03:47:20Z","cross_cats_sorted":[],"title_canon_sha256":"67aaebc4b36bfe1ca769e25eacce2449e60bb0956ae43d2087e8e2938c369035","abstract_canon_sha256":"81afe66f9ca1c02052df39e1bfd99e3728e48e043f1a9425ca1a2fb9b8472d15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:30.459884Z","signature_b64":"JId6zm9CJ6fly7LZ+NMaoEyPlZpP/mFajeXq2baMJesc36l2xYYnRNABkqgIx1ndiWYG/ciAlPB2bYg0ve1VBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f981aed737312e546716e291023a96578aa9c44bdc730fee8f7f9d836aed8db","last_reissued_at":"2026-05-18T04:31:30.459459Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:30.459459Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bounds involving parameter $\\sigma_2$ for the rainbow connection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiuying Dong, Xueliang Li","submitted_at":"2011-01-17T03:47:20Z","abstract_excerpt":"For a graph $G$, we define $\\sigma_2(G)=min \\{d(u)+d(v)| u,v\\in V(G), uv\\not\\in E(G)\\}$, or simply denoted by $\\sigma_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors, which was introduced by Chartrand et al. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow edge-connected. We prove that if $G$ is a connected graph of order $n$, then $rc(G)\\leq 6\\frac{n-2}{\\sigma_2+2}+7$. 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