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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. 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