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

Let Dn,k D n , k be the set of all permutations of the symmetric group Sₙ S n that have no cycles of length i for all 1 ≤ i ≤ k 1 ≤ i ≤ k . In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph Cay(Sₙ,Dn,k) Cay ( S n , D n , k ) is equal to the set of all the largest independent sets in the derangement graph Cay(Sₙ,Dn,1) 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.