### Symmetry Breaking in Graphs

#### Abstract

A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R