  Let $G$ be a simple graph with $n$ vertices and list chromatic number
$\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex
of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1]
conjectured that at least $\frac{tn}{\chi_\ell}$ vertices of $G$ can be colored
from these lists. In this paper we find some new results in partial list
coloring which help us to show that the conjecture is true for at least half of
the numbers of the set $\{1,2,...,\chi_\ell(G)-1\}$. In addition we introduce a
new related conjecture and finally we present some results about this
conjecture.
