Abstract |
Suppose Gamma is a group acting on a set X. An r-labeling phi: X to {1, 2, ..., r} of X is distinguishing (with respect to the action of Gamma) if for any sigma in Gamma, sigma not equal id_X, there exists an element x in X such that phi(x) not equal phi(sigma(x)). The distinguishing number, D_{Gamma}(X), of the action of Gamma on X is the minimum r for which there is an r-labeling which is distinguishing. Given a graph G, the distinguishing number of G, D(G),is defined as D(G) = D_{Aut(G)}(V(G)). This thesis determines the distinguishing numbers of all actions of S_5. As a consequence, we determine all the possible values of the distinguishing numbers of graphs G with Aut(G)=S_5, confirming a conjecture of Albertson and Collins. |