The distinguishing chromatic number of bipartite graphs of girth at least six

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number ${\chi }_D(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $\mathrm{\Delta }(G)$, then ${\chi }_D(G)\le \mathrm{\Delta }(G)+1$. We also obtain an upper bound for ${\chi }_D(G)$ where $G$ is a graph with at most one cycle. Finally, we state a relationship between the distinguishing chromatic number of a graph and its spanning subgraphs.