LIST-3-COLORING ORDERED GRAPHS WITH A FORBIDDEN INDUCED SUBGRAPH

The List-3-Coloring Problem is to decide, given a graph G and a list L(v) \subseteq \{1, 2, 3\} of colors assigned to each vertex v of G, whether G admits a proper coloring \phi with \phi(v) \in L(v) for every vertex v of G, and the 3-Coloring Problem is the List-3-Coloring Problem on instances with L(v) = \{1, 2, 3\} for every vertex v of G. The List-3-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to H-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph H), it remains NP-complete unless H 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 P8-free graphs (where P8 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 G and H, we say G is H-free if H is not isomorphic to an induced subgraph of G with the isomorphism preserving the linear order. We prove, assuming H to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to H-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if H has at most one edge, and remains NP-complete if H has at least three edges. Moreover, in the case where H has exactly two edges, we give a complete dichotomy when the two edges of H share an end, and prove several NP-completeness results when the two edges of H do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.