Bounds on the Distinguishing Chromatic Number

Collins and Trenk define the distinguishing chromatic number chi(D)(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 chi(D)(G) based on the underlying graph G. In this paper we prove results that relate chi(D)(G) to the automorphism group of G. We prove two upper bounds for chi(D)(G) in terms of the chromatic number chi(G) and show that each result is tight : (1) if Aut (G) is any finite group of order p(1)(i1)p(2)(i2) ... p(k)(ik) chi(D)(G) <= chi(G) + i(1) + i(2) ...+ i(k), and (2) if Aut(G)isafinite and abelian group written Aut(G) = Z(p1i1) x ... x Z(pkik) then we get the improved bound chi(D)(G) <= chi(G) + k. In addition, we characterize automorphism groups of graphs wit h chi(D)(G) = 2 and discuss similar results for graphs with chi(D)(G) = 3.