Subgroup perfect codes of S n in Cayley graphs

A perfect code in a graph 𝚪 = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \ C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. In this work, we present a classification of cyclic 2-subgroup perfect codes of S n . We analyze these subgroups, detailing their structure and properties. We extend our discussion to various classes of subgroup perfect codes of the symmetric group S n , encompassing both commutative and non-commutative cases. We provide numerous examples to illustrate and support our findings.