On the Lucky Choice Number of Graphs |
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Authors: | S Akbari M Ghanbari R Manaviyat S Zare |
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Institution: | 1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran 2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran 3. Mathematics Department, Payame Noor University, 19395-4697, Tehran, Iran 4. Department of Mathematical Sciences, Amirkabir University of Technology, Tehran, Iran
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Abstract: | Suppose that G is a graph and ${f: V (G) \rightarrow \mathbb{N}}$ 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 ${S(u) \neq S(v),}$ for every pair of adjacent vertices u and v. Also, for each vertex ${v \in V(G),}$ let L(v) denote a list of natural numbers available at v. A list lucky labeling, is a lucky labeling f such that ${f(v) \in L(v),}$ for each ${v \in V(G).}$ 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, η l (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 ${\Delta \geq 2, \eta_{l}(G) \leq \Delta^2-\Delta + 1,}$ 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. |
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