{"paper":{"title":"A Lower Bound and Several Exact Results on the $d$-Lucky Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Ahima Emilet, Indra Rajasingh, Sandi Klav\\v{z}ar","submitted_at":"2019-03-19T07:15:23Z","abstract_excerpt":"If $\\ell: V(G)\\rightarrow {\\mathbb N}$ is a vertex labeling of a graph $G = (V(G), E(G))$, then the $d$-lucky sum of a vertex $u\\in V(G)$ is $d_\\ell(u) = d_G(u) + \\sum_{v\\in N(u)}\\ell(v)$. The labeling $\\ell$ is a $d$-lucky labeling if $d_\\ell(u)\\neq d_\\ell(v)$ for every $uv\\in E(G)$. The $d$-lucky number $\\eta_{dl}(G)$ of $G$ is the least positive integer $k$ such that $G$ has a $d$-lucky labeling $V(G)\\rightarrow [k]$. A general lower bound on the $d$-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07863","kind":"arxiv","version":1},"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"}