  The computational complexity of the Vertex Coloring problem is known for all
hereditary classes of graphs defined by forbidding two connected five-vertex
induced subgraphs, except for seven cases. We prove the polynomial-time
solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$,
banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.
