Proper Distinguishing Colorings with Few Colors for Graphs with Girth at Least 5

Abstract

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.  Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $\chi_D(G)\leqslant \Delta+1$. We prove this conjecture.