Distance in cayley graphs on permutation groups generated by km-Cycles

In this paperý, ýwe extend upon the results of Bý. ýSuceav{\u{a}} and Rý. ýStong [Amerý. ýMathý. ýMonthlyý, ý110 (2003) 162--162]ý, ýwhich they computed the minimum number of 3-cycles needed to generate an even permutationý.
ýLet Ωnk,m be the set of all permutations of the form c1c2⋯cký ýwhere ci's are arbitrary m-cycles in Sný. ýSuppose that Γnk,m be the Cayley graph on subgroup of Sn generated by all permutationsý ýin Ωnk,mý. ýWe find a shortest path joining identity and any vertex of Γnk,mý, ýfor arbitrary natural number ký, ýand m=2ý,ý\ý,ý3,\ý,ý4ý. ýAlsoý, ýwe calculate the diameter of these Cayley graphsý. ýAs an applicationý, ýwe present an algorithm for finding a short expression of a permutation as products of given permutationsý.