| The M(3) and M(4) property of some complete multipartite graphs are researched in this paper. To begin with, it is proved that none of K1*4,5, K1*4,4, K2,2,4 and K2,2,5 is uniquely 3-list colorable and their m-number are all 3. Last, we characterize almost all uniquely 4-list colorable complete multipartite graphs, which have more than 5 parts, except finitely many of them.
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