In this paper uniquely list colorable graphs are studied. A graph G is said 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 χ ′ ℓ –critical graph is χ ′ – critical and the equivalence of this conjecture to the well known list coloring conjecture is shown. 1

Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that |L(v) | = k for every vertex v and the graph has exactly one L-coloring with these lists. Mahdian and Mahmoodian [MM99] gave a polynomial-time characterization of uniquely 2-list colorable graphs. Answering an open question from [GM01,MM99], we show that uniquely 3-list colorable graphs are unlikely to have such a nice characterization, since recognizing these graphs is Σ p 2-complete. 1

A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used.