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

Abstract

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 ∈ {1, …, k}. The distinguishing chromatic number of a graph G, denoted χ_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 ∈ ℕ, there exists an infinite sequence of vertex-transitive graphs G_i = (V_i, E_i) such that

1. χ_D(G_i) > χ(G_i) > k,

2. |Aut(G_i)| < 2k|V_i|, 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.