Perfect codes of bi-Cayley graphs

A bi-Cayley graph is a graph that has a semiregular group H of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup G such that H is a subgroup of G. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.