DISTANCE IN CAYLEY GRAPHS ON PERMUTATION GROUPS GENERATED BY k m-CYCLES

In this paper, we extend upon the results of B. Suceava and R. Stong [Amer. Math. Monthly, 110 (2003) 162{162], which they computed the minimum number of 3-cycles needed to gener- ate an even permutation. Let Ωnk ;m be the set of all permutations of the form c1c2 ck where ci's are arbitrary m-cycles in Sn. Suppose that 􀀀nk ;m be the Cayley graph on subgroup of Sn generated by all permutations in Ωnk ;m. We nd a shortest path joining identity and any vertex of 􀀀nk ;m, for arbitrary natural number k, and m = 2; 3; 4. Also, we calculate the diameter of these Cayley graphs. As an application, we present an algorithm for nding a short expression of a permutation as products of given permutations.