  This paper studies the on-line choice number on complete multipartite graphs
with independence number $m$. We give a unified strategy for every prescribed
$m$. Our main result leads to several interesting consequences comparable to
known results. (1) If $ k_1-\sum_{p=2}^m(\frac{p^2}{2}-\frac{3p}{2}+1)k_p\geq
0$, where $k_p$ denotes the number of parts of cardinality $p$, then $G$ is
on-line chromatic-choosable. (2) If $
|V(G)|\leq\frac{m^2-m+2}{m^2-3m+4}\chi(G)$, then $G$ is on-line
chromatic-choosable. (3) The on-line choice number of regular complete
multipartite graphs $K_{m\star k}$ is at most $(m+\frac{1}{2}-\sqrt{2m-2})k$
for $m\geq 3$.
