Symmetry Breaking in Graphs

A labeling of the vertices of a graph G, ` : V (G) !f1;:::;rg, 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 i, there is a graph G such that Aut(G) » ia ndD(G )=2 ; (ii)D(G )= O(log(jAut(G)j)); (iii) if Aut(G) is abelian, then D(G) • 2; (iv) if Aut(G) is dihedral, then D(G) • 3; and (v) If Aut(G) » S4 ,t hen either D(G )= 2 orD(G) = 4. Mathematics Subject Classiflcation 05C,20B,20F,68R