$\chi_D(G)$, $|Aut(G)|$, and a variant of the Motion Lemma

TL;DR

This paper introduces a variant of the Motion lemma to analyze the distinguishing chromatic number of graphs, providing new examples and families of graphs with specific automorphism and coloring properties.

Contribution

It proves a new lemma related to the Motion lemma, and constructs examples of graphs with prescribed automorphism and distinguishing chromatic number characteristics.

Findings

Graphs with $ ext{distinguishing chromatic number} = ext{chromatic number} + 1$

Existence of graphs with large automorphism groups where all proper colorings are distinguishing

Families of bipartite graphs with arbitrarily large distinguishing chromatic number

Abstract

The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $\chi_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $\phi$ of the graph $G$ fixes each color class of $G$. In this paper, 1. We prove a lemma that may be considered a variant of the Motion lemma of \cite{RS} and use this to give examples of several families of graphs which satisfy $\chi_D(G)=\chi(G)+1$. 2.We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also describe families of graphs with (relatively) very small automorphism groups which satisfy $\chi_D(G)=\chi(G)+1$, for arbitrarily large values of $\chi(G)$. 3. We describe non-trivial families of bipartite graphs that satisfy $\chi_D(G)>r$ for any positive integer $r$.