  The on-line choice number of a graph is a variation of the choice number
defined through a two person game. It is at least as large as the choice number
for all graphs and is strictly larger for some graphs. In particular, there are
graphs $G$ with $|V(G)| = 2 \chi(G)+1$ whose on-line choice numbers are larger
than their chromatic numbers, in contrast to a recently confirmed conjecture of
Ohba that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ has its choice number
equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture
was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method
to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$
with $|V(G)| \le 2 \chi(G)$ has its on-line choice number equal its chromatic
number. This paper confirms the on-line version of Ohba conjecture for graphs
$G$ with independence number at most 3. We also study list colouring of
complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove
that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and
present an alternate proof of Kierstead's result that its choice number is
$\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le
\chi(G)+\sqrt{\chi(G)}$ then its on-line choice number equals chromatic number.
