A note on “Largest independent sets of certain regular subgraphs of the derangement graph”

Let \(D_{n,k}\) be the set of all permutations of the symmetric group \(S_n\) that have no cycles of length i for all \(1 \le i \le k\). In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph \(\text {Cay}(S_n,D_{n,k})\) is equal to the set of all the largest independent sets in the derangement graph \(\text {Cay}(S_n,D_{n,1})\), provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group.