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

.The List-3-Coloring Problem is to decide, given a graph \(G\) and a list \(L(v)⊆ \{1,2,3\}\) of colors assigned to each vertex \(v\) of \(G\), whether \(G\) admits a proper coloring \(φ\) with \(φ (v)∈ L(v)\) 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 \(P_8\)-free graphs (where \(P_8\) 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.