Four-coloring p6-free graphs. i. extending an excellent precoloring

This is the first paper in a series whose goal is to give a polynomial-time algorithm 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.