Enumeration of hamiltonian paths in Cayley diagrams

LetG be a group generated by a subset of elementsS. The Cayley diagram ofG givenS is the labeled directed graph with vertices identified with the elements ofG and (v, u) is an edge labeledh ifh S anduh=v. The sequence of elements ofS corresponding to the edges transversed in a hamiltonian path (whose initial vertex is the identity) is called a group generating sequence (abbreviatedggs) inS.In this paper a minimal upper bound for the number ofggs's in a pair of generator elements for any two-generated group is given. For all groups of the formG=a, b:b ⁿ =1,a ^m =b ^r ,ba=ab ^–1 wherem is even, it is shown that the number ofggs's in {a, b} is 1+m(n–1)/2. An algorithm is developed that yields the number ofggs's for two-generated groupsG=a, b for which ba ^–1G. Explicit forms for the countedggs's are also provided.