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

Document Type : Research Paper

Authors

Department Mathematics, Yazd University 89195-741, Yazd, Iran

Abstract

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.

Keywords

#### References

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