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 frac(2, 3) k l ≤ n (k, l) ≤ frac(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].