On the Lucky Choice Number of Graphs

Suppose that G is a graph and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f: V (G) \rightarrow \mathbb{N}}$$\end{document} is a labeling of the vertices of G. Let S(v) denote the sum of labels over all neighbors of the vertex v in G. A labeling f of G is called lucky if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S(u) \neq S(v),}$$\end{document} for every pair of adjacent vertices u and v. Also, for each vertex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v \in V(G),}$$\end{document} let L(v) denote a list of natural numbers available at v. A list lucky labeling, is a lucky labeling f such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(v) \in L(v),}$$\end{document} for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v \in V(G).}$$\end{document} A graph G is said to be lucky k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list lucky labeling of G. The lucky choice number of G, ηl(G), is the minimum natural number k such that G is lucky k-choosable. In this paper, we prove that for every graph G with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta \geq 2, \eta_{l}(G) \leq \Delta^2-\Delta + 1,}$$\end{document} where Δ denotes the maximum degree of G. Among other results we show that for every 3-list assignment to the vertices of a forest, there is a list lucky labeling which is a proper vertex coloring too.