  If the vertices of a graph $G$ are colored with $k$ colors such that no
adjacent vertices receive the same color and the sizes of any two color classes
differ by at most one, then $G$ is said to be equitably $k$-colorable. Let
$|G|$ denote the number of vertices of $G$ and $\Delta=\Delta(G)$ the maximum
degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is
equitably $\Delta$-colorable if $G$ satisfies $(|G|+1)/3 \leq \Delta < |G|/2$
and none of its components is a $K_{\Delta +1}$.
