TY - JOUR ID - 1061 TI - The distinguishing chromatic number of bipartite graphs of girth at least six JO - Algebraic Structures and Their Applications JA - AS LA - en SN - 2382-9761 AU - Alikhani, Saeid AU - Soltani, Samaneh AD - Department Mathematics, Yazd University 89195-741, Yazd, Iran Y1 - 2016 PY - 2016 VL - 3 IS - 2 SP - 81 EP - 87 KW - distinguishing number KW - distinguishing chromatic number KW - symmetry breaking DO - N2 - 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. UR - https://as.yazd.ac.ir/article_1061.html L1 - https://as.yazd.ac.ir/article_1061_d7a2c4d97e197bfadafec3fd409da617.pdf ER -