Symmetry Breaking in Graphs

A labeling of the vertices of a graph G, φ : V (G) → {1, . . . , 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 Γ, there is a graph G such that Aut(G) ∼= Γ and D(G) = 2; (ii) D(G) = O(log(|Aut(G)|)); (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, then either D(G) = 2 or D(G) = 4. Mathematics Subject Classification 05C,20B,20F,68R