Author:
Filmus Yuval,Lindzey Nathan
Abstract
AbstractLet $$D_{n,k}$$
D
n
,
k
be the set of all permutations of the symmetric group $$S_n$$
S
n
that have no cycles of length i for all $$1 \le i \le k$$
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 $$\text {Cay}(S_n,D_{n,k})$$
Cay
(
S
n
,
D
n
,
k
)
is equal to the set of all the largest independent sets in the derangement graph $$\text {Cay}(S_n,D_{n,1})$$
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.
Funder
HORIZON EUROPE European Innovation Council
Technion - Israel Institute of Technology
Publisher
Springer Science and Business Media LLC
Reference8 articles.
1. Ahmadi, B.: Maximum intersecting families of permutations. PhD thesis, University of Regina (2013)
2. Ahmadi, B., Meagher, K.: A new proof for the Erdos–Ko–Rado theorem for the alternating group. Discrete Math. 324, 28–40 (2014)
3. Dafni, N., Filmus, Y., Lifshitz, N., Lindzey, N., Vinyals, M.: Complexity measures on the symmetric group and beyond (extended abstract). In: Lee, J.R., (ed.) 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6–8, 2021, Virtual Conference, Volume 185 of LIPIcs, pp. 87:1–87:5. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2021)
4. Ellis, D., Friedgut, E., Pilpel, H.: Intersecting families of permutations. J. Amer. Math. Soc. 24, 649–682 (2011)
5. Ku, C.Y., Lau, T., Wong, K.B.: Largest independent sets of certain regular subgraphs of the derangement graph. J. Algebraic Combin. 44, 81–98 (2016)