Covering the alternating groups by products of cycle classes

Given integers k , l ⩾ 2 , where either l is odd or k is even, we denote by n = n ( k , l ) the largest integer such that each element of A n is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay ( A n , C l ) , where C l is the set of all l-cycles in A n . We prove that if k ⩾ 2 and l ⩾ 9 is odd and divisible by 3, then 2 3 k l ⩽ n ( k , l ) ⩽ 2 3 k l + 1 . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368–380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87–99].