  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 $P_6$-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.
