2020/06/19 by Zhang, Junyang, Zhou, Sanming
#05C25 #05C69 #94B25 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.11104
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. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing the identity element such that (A, H) is a tiling of G in the sense that every element of G can be uniquely expressed as the product of an element of A and an element of H. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving 2-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and 2-groups.