Distance in cayley graphs on permutation groups generated by $k$ $m$-Cycles

In this paper, we extend upon the results of B. Suceavă and R. Stong [Amer. Math. Monthly, 110 (2003) 162–162], which they computed the minimum number of 3-cycles needed to generate an even permutation. Let Ωk,m be the set of all permutations of the form c1c2 · · · ck where ci’s are arbitrary m-cycles in Sn. Suppose that Γ n k,m be the Cayley graph on subgroup of Sn generated by all permutations in Ωk,m. We find a shortest path joining identity and any vertex of Γ n k,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 finding a short expression of a permutation as products of given permutations.