  The list coloring problem is a variation of the classical vertex coloring
problem, extensively studied in recent years, where each vertex has a
restricted list of allowed colors, and having some variations as the
$(\gamma,\mu)$-coloring, where the color lists have sequential values with
known lower and upper bounds. This work discusses the choosability property,
that consists in determining the least number $k$ for which it has a proper
list coloring no matter how one assigns a list of $k$ colors to each vertex.
This is a $\Pi_2^P$-complete problem, however, we show that
$k$-$(\gamma,\mu)$-choosability is an $NP$-problem due to its relation with the
$k$-coloring of a graph and application of methods of proof in choosability for
some classes of graphs, such as complete bipartite graph, which is $ 3
$-choosable, but $ 2 $-$(\gamma,\mu)$-choosable.
