Subgroup Perfect Codes in Bi-Cayley Graphs

In this paper, we study perfect codes in bi-Cayley graphs, focusing on when two subgroups $H$ and $K$ of a group G can form a perfect code of the form $H_0\cup K_1$. We first consider the problem for a general group G and then examine what happens for special classes of groups, particularly abelian groups. Along the way, we obtain several results, including a necessary and sufficient condition for normal subgroups to form a perfect code, as well as an equivalent condition for abelian groups in terms of their Sylow 2-subgroups. Finally, we study the same problem for dihedral groups, providing results for this important class of non-abelian groups.