Note on property M(k) of some complete multipartite graphs

The property M(k) is a concept associated with the unique list coloring. A graph G has the property M(k) if for any collection of lists assigned to its vertices, each of size k, either there is no list coloring for G or there are at least two k-list colorings. The existing research results indicate that K1,1,1,r and K1⁎r,3 have the property M(3), and in addition K1⁎5,r and K1⁎r,5 have the property M(4) for every r≥2. The results above are extended to every k in this paper. We will show that for every r≥1, k≥2, K1⁎r,(2k−3) and K1⁎(2k−3),r have the property M(k). The conclusion will pave the way to characterizing unique k-list colorable complete multipartite graphs.