List 3-Coloring on Comb-Convex and Caterpillar-Convex Bipartite Graphs

Given a graph $G=(V,E)$ and a list of available colors $L(v)$ for each vertex $v\in V$, where $L(v)\subseteq \{1,2,\dots ,k\}$, 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.