K 1,p k-factorization of complete bipartite graphs

In this paper, it is shown that a sufficient condition for the existence of a K ₁,p^k-factorization of K _m,n ⁹ whenever p is a prime number and k is a positive integer, is (1) m<-p ^k n,(2)n <-p ^k m,(3)p ^k n-m≡p ^k m-n≡(mod (p ^2k −1)) and (4)(p ^k n-m)(p ^k m-n)≡0(mod (p ^k -1)p ^k x(p ^2k -1) (m+n)).