The basis number of the powers of the complete graph

A basis of the cycle space C(G) of a graph G is h-fold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that C(G) has an h-fold basis. MacLane [3] showed that a graph G is planar if and only if b(G) ≤ 2. Schmeichel [4] proved that b(Kn) ≤ 3, and Banks and Schmeichel [2] proved that b(Kd 2 ) ≤ 4 where Kd 2 is the d-dimesional hypercube. We show that b(Kd n) ≤ 9 for any n and d, where Kd n is the cartesian d-th power of the complete graph Kn.