Some concepts in list coloring

Abstract

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 vertex 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 $\chi_\ell$-critical graph is $\chi'$-critical and the equivalence of this conjecture to the well known list coloring conjecture is shown.