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. 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 \ (\) with \ ( (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 \ (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. Keywordscoloringlist-coloringalgorithminduced subgraphordered graphcomputational complexityMSC codes05C1505C85
Hajebi et al. (Wed,) studied this question.