Distinguishing coloring

In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.

Distinguishing colorings and distinguishing numbers were introduced by Albertson & Collins (1996), who provided the following motivating example, based on a puzzle previously formulated by Frank Rubin: "Suppose you have a ring of keys to different doors; each key only opens one door, but they all look indistinguishable to you. How few colors do you need, in order to color the handles of the keys in such a way that you can uniquely identify each key?" This example is solved by using a distinguishing coloring for a cycle graph. With such a coloring, each key will be uniquely identified by its color and the sequence of colors surrounding it.