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Some concepts in list coloring

Ch. Eslahchi·M. Ghebleh·H. Hajiabolhassan
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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 $χ_\ell$-critical graph is $χ'$-critical and the equivalence of this conjecture to the well known list coloring conjecture is shown.

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Ch. Eslahchi, M. Ghebleh, H. Hajiabolhassan. 2008-01-02. Some concepts in list coloring. https://arxiv.org/abs/math/9906011

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