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The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\\lbrace 1, 2, ...,k\\rbrace$ is the lucky number of $G$, denoted by $\\eta(G)$. We will prove, for every graph $G$ other than $ K_{2} $, $\\frac{w}{n-w+1}\\leq\\eta (G) \\leq \\Delta^{2} $ and we present an algorithm for l"},"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":"1007.2480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-15T05:56:55Z","cross_cats_sorted":[],"title_canon_sha256":"d06f74482fea209c7cebf6e9e3bb9aa76915f26ae14d1cbeecf9f3d9e1653139","abstract_canon_sha256":"f7a49f5f4c68d8132c43bb13797e0f0869ac08c8806a6d6c89f9cc2f8ee61691"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:50.341410Z","signature_b64":"PC4lVuj2s+k5IIFDKyA58975akagM2ZRmDV1My0z6R3/+m+5rTunJxFBg1Py9HkpkIcKR1eMj6lxhJPCjSXPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c38e20634162e6d06f89ad75c7998f5f8f3e7ef3cfd41123518a336840ab253b","last_reissued_at":"2026-05-18T04:38:50.340936Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:50.340936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Lucky labeling of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ali Dehghan, Arash Ahadi, Esmael Mollaahmadi","submitted_at":"2010-07-15T05:56:55Z","abstract_excerpt":"Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. 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