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In this paper, it is shown that a sufficient condition for the existence of a K1,pk-factorization of Km,n, whenever p is a prime number and k is a positive integer, is (1) m≤pkn,(2) n≤pkm,(3)pkn-m≡pkm- n≡0(mod(p2k - 1)) and (4)(pkn-m)(pkm- n)≡0(mod(pk - 1)pk×(p2k-1)(m+n)).