In this paper, all results apply only to finite graphs. Let G be a simple connected finite graph with n vertices and maximum degree \(\Delta (G)\) . We show that the list-distinguishing chromatic number \(\chi _{D_{L}}(G)\) of G is at most \(2\Delta (G)\) , and it is \(2\Delta (G)\) if G is a complete bipartite graph \(K_{\Delta (G),\Delta (G)}\) or a cycle with six vertices. We apply a result of Lovász to reduce the above-mentioned upper bound of \(\chi _{D_{L}}(G)\) for certain graphs. We also show that if H is a connected unicyclic graph of girth of at least seven and \(\Delta (H)\ge 3\) , then \(\chi _{D_{L}}(H)\) is at most \(\Delta (H)\) . Moreover, we obtain two upper bounds for \(\chi _{D_{L}}(G)\) in terms of the coloring number of G and the list chromatic number of G. We also determine the list-distinguishing chromatic number for some special graphs.
