4-coloring (P6,bull)-free graphs

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 (P6,bull,gem)-free graphs.