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The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected.\n  In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\\frac{\\log n+\\om}{n}$ where $\\om=\\om(n)\\to\\infty$ and ${\\om}=o(\\log{n})$ and of random $r$-regular graphs where $r \\geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)"},"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":"1201.4603","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-22T21:22:33Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"f4b3541bf108191a28ac334a28fb576ac7e197ef7828c2298b94aeff6ddc833f","abstract_canon_sha256":"0c87dd7287fcd754836197f2ea170f72ea76eceb358d5352d551e324392e5c69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:15.428739Z","signature_b64":"VhrNW2uv9HFF8jl8ZyKMbfFoYiKyx97f+IVJkXmzoVyDBecU62mBffPq+Em2Ul2AY6QFJwpWVFYloKaOAw23AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"031dc53f16120ec60202a5213cf3b75305d8981a3dc0fdddd619bbba5c0a90ee","last_reissued_at":"2026-05-18T03:44:15.428338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:15.428338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rainbow Connectivity of Sparse Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Alan Frieze, Charalampos E. Tsourakakis","submitted_at":"2012-01-22T21:22:33Z","abstract_excerpt":"An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected.\n  In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\\frac{\\log n+\\om}{n}$ where $\\om=\\om(n)\\to\\infty$ and ${\\om}=o(\\log{n})$ and of random $r$-regular graphs where $r \\geq 3$ is a fixed integer. 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