|
This article is cited in 1 scientific paper (total in 1 paper)
Discrete mathematics and mathematical cybernetics
Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs
I. Yu. Mogilnykhab a Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia
b Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
Abstract:
The Star graph $S_n$ is the Cayley graph of the symmetric group $\mathrm{Sym}_n$ with the generating set $\{(1\ i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi\in \mathrm{Sym}_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n$, $n\geq 3$. In this note we show that for any $n$, $n\geq 6$ the Star graph $S_n$ contains a perfect code which is the union of cosets of the embedding of $\mathrm{PGL}(2,5)$ into $\mathrm{Sym}_6$.
Keywords:
perfect code, efficient dominating set, Cayley graph, Star graph, projective linear group, symmetric group.
Received December 4, 2019, published April 10, 2020
Citation:
I. Yu. Mogilnykh, “Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs”, Sib. Èlektron. Mat. Izv., 17 (2020), 534–539
Linking options:
https://www.mathnet.ru/eng/semr1229 https://www.mathnet.ru/eng/semr/v17/p534
|
Statistics & downloads: |
Abstract page: | 193 | Full-text PDF : | 103 | References: | 12 |
|