Most Regular Graphs Are Quickly R-transitive

Consider a d-regular directed graph, G, on n vertices determined by d permutations 1; : : : ; d 2 Sn. Each word, w, over the alphabet = f 1; 2; : : : ; dg determines a walk at each vertex, and these walks give rise to a permutation of the vertices (which is simply the product of the permutations). Such a word maps a distinct r-tuple of vertices, (v1; : : : ; vr) to another distinct r-tuple of vertices, (u1; : : : ; ur). We say that G is r-transitive if for every two distinct r-tuples (v1; : : : ; vr) and (u1; : : : ; ur) there is a word, w, taking the vi to the ui; furthermore we say that G is c-quickly r-transitive if there is always a word, w, of length c logn, acheiving this. In this paper we prove that for every r and d 2 there is a c such that most graphs, G, are c-quickly r-transitive, then the number n of vertices becomes large. Although we came across this problem while studying a rather unrelated cryptographic problem (which will be discussed in the paper), it belongs to a general context where random Cayley graphs are proved good expanders: note that, our theorem says that for xed r the quotients Sn=Sn r of most Cayley graphs on Sn are good expanders.