Perfect packings with complete graphs minus an edge

Let K r − denote the graph obtained from K r by deleting one edge. We show that for every integer r ≥ 4 there exists an integer n 0 = n 0 ( r ) such that every graph G whose order n ≥ n 0 is divisible by r and whose minimum degree is at least ( 1 − 1 / χ c r ( K r − ) ) n contains a perfect K r − -packing, i.e. a collection of disjoint copies of K r − which covers all vertices of G . Here χ c r ( K r − ) = r ( r − 2 ) r − 1 is the critical chromatic number of K r − . The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n .