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    $\begingroup$ But, there are planar graphs that are not 4-choosable (incidentally, I think the smallest known example is due to recent Fields' medalist Maryam Mirzakhani), so this generalization is false. $\endgroup$ Commented Dec 8, 2014 at 1:14
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    $\begingroup$ ...who found it in 1996, when she was 19 years old. $\endgroup$ Commented Dec 8, 2014 at 8:54
  • $\begingroup$ well, it was not meant as a generalization. while it is known that the chromatic number gives a lower bound for the choosability number, the choosability number cannot be bounded from above by a function of the chromatic number, so they are indeed two different objects. i mentioned thomassen's theorem simply as a related result. $\endgroup$ Commented Dec 8, 2014 at 8:59