  Given a graph $G=(V, E)$ and a list of available colors $L(v)$ for each
vertex $v\in V$, where $L(v) \subseteq \{1, 2, \ldots, k\}$, List $k$-Coloring
refers to the problem of assigning colors to the vertices of $G$ so that each
vertex receives a color from its own list and no two neighboring vertices
receive the same color. The decision version of the problem List $3$-Coloring
is NP-complete even for bipartite graphs, and its complexity on comb-convex
bipartite graphs has been an open problem. We give a polynomial-time algorithm
to solve List $3$-Coloring for caterpillar-convex bipartite graphs, a
superclass of comb-convex bipartite graphs. We also give a polynomial-time
recognition algorithm for the class of caterpillar-convex bipartite graphs.
