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This is the first paper in a series whose goal is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial-time algorithm that determines if a special kind of precoloring of a P6-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]