  Let $L$ be a set of graphs. $Free$($L$) is the set of graphs that do not
contain any graph in $L$ as an induced subgraph. It is known that if $L$ is a
set of four-vertex graphs, then the complexity of the coloring problem for
$Free$($L$) is known with three exceptions: $L $= {claw, $4K_1$}, $L$ = {claw,
$4K_1$, co-diamond}, and $L$ = {$C_4$, $4K_1$}. In this paper, we study the
coloring problem for $Free$(claw, $4K_1$). We solve the coloring problem for a
subclass of $Free$(claw, $4K_1$) which contains the class of $4K_1$-free line
graphs. Our result implies the chromatic index of a graph with no matching of
size four can be computed in polynomial time.
