A question about permutation groups: I wonder if someone who is expert in permutation group theory could answer the following question.

Let $x \in S_n$ (the symmetric group) be an involution which is the product of k disjoint transpositions.

For any permutation $y \in S_n$, let f(y) be the length of a
shortest **reduced** word in $x, y, \mbox{ and } y^{-1}$ that equals
the identity.

Finally, let F(k,n) be the maximum of f(y) over all (non-involutions) $y \in S_n$ ($n >= 2k$).

Question: Is there an easy way to determine F(k,n)?

For (very) small k the evidence suggests something like F(k,n) = 12k. For k=1 this is easy to show by a case analysis, based on how the transposition intersects the cycles of y. Even for k=2 I found the problem harder than expected. Perhaps this has been studied before?

Of course this could be restated as a question about the maximum girth of Type II, 3-regular Cayley graphs on $S_n$.