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. 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.