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 $f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial automorphism of $G$ preserves the sets $f^{-1}(i)$ for each $i\in\{1,\ldots,k\}$. 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 $k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs $G_{i}=(V_i,E_i)$ such that $χ_D(G_i)>χ(G_i)>k$ and $|\mathrm{Aut}(G_i)|=O_k(|V_i|)$, where $\mathrm{Aut}(G_i)$ denotes the full automorphism group of $G_i$. In particular, this answers a problem raised in the paper $χ_D(G)$, $|\mathrm{Aut}(G)|$ and a variant of the Motion lemma.