Geodesic
Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 534-539

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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. 
@article{SEMR_2020_17_a67,
     author = {I. Yu. Mogilnykh},
     title = {Perfect codes from $\mathrm{PGL}(2,5)$ in {Star} graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {534--539},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/}
}
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I. Yu. Mogilnykh. Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 534-539. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a67/