List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1158-1190, March 2024.
Abstract. The List-3-Coloring Problem is to decide, given a graph [math] and a list [math] of colors assigned to each vertex [math] of [math], whether [math] admits a proper coloring [math] with [math] for every vertex [math] of [math], and the 3-Coloring Problem is the List-3-Coloring Problem on instances with [math] for every vertex [math] of [math]. The List-3-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to [math]-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph [math]), it remains NP-complete unless [math] is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the 3-Coloring Problem on [math]-free graphs (where [math] denotes the eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs [math] and [math], we say [math] is [math]-free if [math] is not isomorphic to an induced subgraph of [math] with the isomorphism preserving the linear order. We prove, assuming [math] to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to [math]-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if [math] has at most one edge, and remains NP-complete if [math] has at least three edges. Moreover, in the case where [math] has exactly two edges, we give a complete dichotomy when the two edges of [math] share an end, and prove several NP-completeness results when the two edges of [math] do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.