Graphs with Large Distinguishing Chromatic Number

Michael Cavers, Karen Seyffarth

Abstract


The distinguishing chromatic number $\chi_D(G)$ of a graph $G$ is the minimum number of colours required to properly colour the vertices of $G$ so that the only automorphism of $G$ that preserves colours is the identity. For a graph $G$ of order $n$, it is clear that $1\leq\chi_D(G)\leq n$, and it has been shown that $\chi_D(G)=n$ if and only if $G$ is a complete multipartite graph. This paper characterizes the graphs $G$ of order $n$ satisfying $\chi_D(G)=n-1$ or $\chi_D(G)=n-2$.

Keywords


Distinguishing chromatic number; Distinguishing number; Graph Colouring; Graph automorphism

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