Perfect codes in circulant graphs

A perfect code in a graph Γ = (V,E) is a subset C of V that is an independent set such that every vertex in V \ C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of V is adjacent to exactly one vertex in C. A perfect code in the Hamming graph H(n, q) agrees with a q-ary perfect 1-code of length n in the classical setting. A necessary and sufficient condition for a circulant graph of degree p − 1 to admit a perfect code is given in this talk, where p is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order n and degree p − 1 to have a perfect code, where p is a prime and p the largest power of p dividing n. Similar results for total perfect codes are also obtained.