When $F$ is a set of four-vertex graphs the complexity of the vertex coloring problem in the class of $F$-free graphs is known, with three exceptions: $F = \{\text{claw}, 4K_1\}$, $F = \{\text{claw}, 4K_1, \text{co-diamond}\}$, and $F = \{C_4, 4K_1\}$. We study the coloring problem for $\{\text{claw},4K_1\}$-free graphs. We solve the vertex coloring problem for a subclass of this class which contains the class of $4K_1$-free line graphs. Our result implies that the chromatic index of a graph with no matching of size four can be computed in polynomial time.