The distinguishing chromatic number (Q815210)

Summary: In this paper we define and study the distinguishing chromatic number, \(\chi_D(G)\), of a graph \(G\), building on the work of \textit{M. O. Albertson} and \textit{K. L. Collins} [Electron. J. Comb. 3, Research paper R18 (1996); printed version J. Comb. 3, 259--275 (1996; Zbl 0851.05088)] who studied the distinguishing number. We find \(\chi_D(G)\) for various families of graphs and characterize those graphs with \(\chi_D(G) = |V(G)|\), and those trees with the maximum chromatic distingushing 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.