On the List Distinguishing Number of Graphs

A graph G is said to be k-distinguishable if every vertex of the graph can be colored from a set of k colors such that no non-trivial auto- morphism xes every color class. The distinguishing number D(G) is the least integer k for which G is k-distinguishable. A list assignment to G is an assignment L = fL(v)g v2V (G) of lists of labels to the vertices of G. A distinguishing L-labeling of G is a distinguishing labeling of G where the label of each vertex v comes from L(v). The list distinguishing number of G, Dl(G) is the minimum k such that every list assignment to G in which jL(v)j = k for all v 2 V (G) yields a distinguishing L-labeling of G. In this paper, we study and compute the list-distinguishing number of some families of graphs. We also study graphs with the distinguishing number equal the list distinguishing number.