{"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 $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is, if we have $Sum(u) \\neq Sum(v)$ whenever $u$ and $v$ are adjacent. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2480","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}