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.

翻译：给定图$G=(V, E)$及每个顶点$v\in V$的可用颜色列表$L(v)$，其中$L(v) \subseteq \{1, 2, \ldots, k\}$，列表$k$-染色问题是指为$G$的顶点分配颜色，使得每个顶点从其自身列表中获得颜色，且相邻顶点颜色不同。该问题的判定版本——列表$3$-染色问题——即使在二分图上也是NP完全的，而在梳状凸二分图上的复杂度此前一直是一个开放问题。本文给出了毛虫凸二分图（梳状凸二分图的超类）上列表$3$-染色问题的多项式时间算法，同时提出了毛虫凸二分图类的多项式时间识别算法。