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Paper IPM / M / 496  


Abstract:  
In this paper uniquely list colorable graphs are studied. A Graph
G is called to be uniquely klist colorable if it admits a
klist assignment from which G has a unique list coloring. The
minimum k for which G is not uniquely klist colorable is
called the mnumber of G. We show that every trianglefree
uniquely colorable graph with chromatic number k+1, is uniquely
klist colorable. A bound for the mnumber of graphs is given,
and using this bound it is shown that every planar graph has
mnumber at most 4. Also we introduce list criticality in graphs
and characterize all 3list 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.
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