A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the colors is the identity map. It is proper-distinguishing if the coloring is both proper and distinguishing. The distinguishing number of G, D(G), is the smallest integer k so that G has a distinguishing k-coloring; the distinguishing chromatic number of G, χD(G), is defined similarly. It has been sh...

Let G = (V (G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k−color set. A (proper) total k−coloring f of G is a function f : T (G) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6=...

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d labels that is preserved only by a trivial automorphism. We prove that Cartesian products of relatively prime graphs whose sizes do not differ too much can be distinguished with a small number of colors. We determine the distinguishing number of the Cartesian product Kk ¤Kn for all k and n, eith...

A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, χD(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring....

A coloring of the vertices of a graph G is said to be distinguishing provided that no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, denoted D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing number, first introduced by Albertson and Collins in 1996, has been widely studied and a number of interesting res...

Nordhaus and Gaddum proved, for any graph G, that χ(G) + χ(G) 6 n + 1, where χ is the chromatic number and n = |V (G)|. Finck characterized the class of graphs, which we call NG-graphs, that satisfy equality in this bound. In this paper, we provide a new characterization of NG-graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of ...

The distinguishing chromatic number χD (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(·). Our main result is that for every graph G, th...