Three-coloring graphs with no induced seven-vertex path II : using a triangle

In this paper we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. This is the second paper in a series of two. The first one, [3] is also submitted to this journal. In [2, 3], a polynomial time algorithm is given for three-coloring triangle-free graphs with no induced sevenvertex path. Combined, this shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time, thus answering a question of [13].