The distinguishing chromatic number

Abstract

In this paper we define and study the distinguishing chromatic number, ΧD(G), of a graph G, building on the work of Albertson and Collins who studied the distinguishing number. We find ΧD(G) for various families of graphs and characterize those graphs with ΧD(G) = |V(G)|, and those trees with the maximum chromatic distinguishing number for trees. We prove analogs of Brooks' Theorem for both the distinguishing number and the distinguishing chromatic number, and for both trees and connected graphs. We conclude with some conjectures.