Upper bounds for the list-distinguishing chromatic number

Abstract: We prove analogs of Brooks' Theorem for the list-distinguishing chromatic number of different classes of simple finite connected graphs. Moreover, we determine two upper bounds for the list-distinguishing chromatic number of a graph G in terms of the coloring number of G and the list-chromatic number of G. We also determine the list-distinguishing chromatic number for various families of graphs (for example: the book graphs).