A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the...

The distinguishing number of a graph G is the smallest positive integer r such that G has a labeling of its vertices with r labels for which there is no non-trivial automorphism of G preserving these labels. In early work, Michael Albertson and Karen Collins computed the distinguishing number for various finite graphs, and more recently Wilfried Imrich, Sandi Klavžar and Vladimir Trofimov compu...

A labeling f : V (G) → {1, 2, . . . , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let χ(G) be the chromatic nu...