# Vertex transitive graphs $G$ with $\chi_D(G) > \chi(G)$ and small automorphism group

## TL;DR

This paper constructs infinite sequences of vertex-transitive graphs where the distinguishing chromatic number exceeds the chromatic number, yet the automorphism group remains small, addressing a key open problem in graph symmetry and coloring.

## Contribution

It proves the existence of vertex-transitive graphs with high distinguishing chromatic number relative to their chromatic number and small automorphism groups, solving an open problem.

## Key findings

- Existence of infinite sequences of such graphs.
- Distinguishing chromatic number can be arbitrarily larger than the chromatic number.
- Automorphism groups of these graphs are linearly bounded by the number of vertices.

## 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 $\chi_D(G)$, 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 $k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs $G_{i}=(V_i,E_i)$ such that $\chi_D(G_i)>\chi(G_i)>k$ 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 $\chi_D(G)$, $|\mathrm{Aut}(G)|$ and a variant of the Motion lemma.