Bounds on the Distinguishing Chromatic Number

Collins and Trenk define the distinguishing chromatic number χD(G) of a graph G to be the minimum number of colors needed to properly color the vertices of G so that the only automorphism of G that preserves colors is the identity. They prove results about χD(G) based on the underlying graphG. In this paper we prove results that relate χD(G) to the automorphism group of G. We prove two upper bounds for χD(G) in terms of the chromatic number χ(G) and show that each result is tight: (1) if Aut(G) is any finite group of order p1 1 p i2 2 · · · p ik k then χD(G) ≤ χ(G)+i1+i2 · · ·+ik, and (2) if Aut(G) is a finite and abelian group written Aut(G) = Z p i1 1 × · · · × Z p ik k then we get the improved bound χD(G) ≤ χ(G) + k. In addition, we characterize automorphism groups of graphs with χD(G) = 2 and discuss similar results for graphs with χD(G) = 3.