%0 Journal Article %T The distinguishing chromatic number of bipartite graphs of girth at least six %J Algebraic Structures and Their Applications %I Yazd University %Z 2382-9761 %A Alikhani, Saeid %A Soltani, Samaneh %D 2016 %\ 11/01/2016 %V 3 %N 2 %P 81-87 %! The distinguishing chromatic number of bipartite graphs of girth at least six %K distinguishing number %K distinguishing chromatic number %K symmetry breaking %R %X 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. We prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $\Delta (G)$,  then    $\chi_{D}(G)\leq \Delta (G)+1$.  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. %U https://as.yazd.ac.ir/article_1061_d7a2c4d97e197bfadafec3fd409da617.pdf