Vertex transitive graphs G with χ D ( G ) > χ( G ) and small automorphism group

For a graph G and a positive integer k, a vertex labelling f : V (G) -> {1, 2,...,k} is said to be k-distinguishing if no non-trivial automorphism of G preserves the sets f(-1) (i) for each i epsilon {1,...,k}. The distinguishing chromatic number of a graph G, denoted chi(D)(G), is defined as the minimum k such that there is a k-distinguishing labelling of V (G) which is also a proper coloring of the vertices of G. In this paper, we prove the following theorem: Given k epsilon N, there exists an infinite sequence of vertex-transitive graphs G(i) = (V-i, E-i) such that 1. chi(D)(G(i)) > chi(G(i)) > k, 2. vertical bar Aut(G(i))vertical bar < 2k vertical bar V-i vertical bar, where Aut(G(i)) denotes the full automorphism group of G(i). In particular, this answers a question posed by the first and second authors of this paper.