On the Lucky Choice Number of Graphs

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, ? 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.