  We say that a graph $G$ is chromatic-choosable when its list chromatic number
$\chi_{\ell}(G)$ is equal to its chromatic number $\chi(G)$.
Chromatic-choosability is a well-studied topic, and in fact, some of the most
famous results and conjectures related to list coloring involve
chromatic-choosability. In 2002 Ohba showed that for any graph $G$ there is an
$N \in \mathbb{N}$ such that the join of $G$ and a complete graph on at least
$N$ vertices is chromatic-choosable. The Ohba number of $G$ is the smallest
such $N$. In 2014, Noel suggested studying the Ohba number, $\tau_{0}(a,b)$, of
complete bipartite graphs with partite sets of size $a$ and $b$. In this paper
we improve a 2009 result of Allagan by showing that $\tau_{0}(2,b) = \lfloor
\sqrt{b} \rfloor - 1$ for all $b \geq 2$, and we show that for $a \geq 2$,
$\tau_{0}(a,b) = \Omega( \sqrt{b} )$ as $b \rightarrow \infty$. We also
initiate the study of some relaxed versions of the Ohba number of a graph which
we call generalized Ohba numbers. We present some upper and lower bounds of
generalized Ohba numbers of complete bipartite graphs while also posing some
questions.
