  We present a polynomial-time algorithm that determines whether a graph that
contains no induced path on six vertices and no bull (the graph with vertices
a, b, c, d, e and edges ab, bc, cd, be, ce) is 4-colorable. We also show that
for any fixed k the k-coloring problem can be solved in polynomial time in the
class of ($P_6$, bull, gem)-free graphs.
