  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 $\chi_D(G)\le \Delta+1$. We prove this
conjecture.
