\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In this paper uniquely list colorable graphs are studied. A Graph
$G$ is called to be uniquely $k$-list colorable if it admits a
$k$-list assignment from which $G$ has a unique list coloring. The
minimum $k$ for which $G$ is not uniquely $k$-list colorable is
called the m-number of $G$. We show that every triangle-free
uniquely colorable graph with chromatic number $k+1$, is uniquely
$k$-list colorable. A bound for the m-number of graphs is given,
and using this bound it is shown that every planar graph has
m-number at most 4. Also we introduce list criticality in graphs
and characterize all 3-list critical graphs. It is conjectured
that every $\chi'_l$-critical graph is $\chi'$-critical and the
equivalence of this conjecture to the well known list coloring
conjecture is shown.
\end{document}