Equitable Coloring of Graphs with Intermediate Maximum Degree

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 $\mathrm{\mid }G\mathrm{\mid }$ denote the number of vertices of $G$ and $\mathrm{\Delta }=\mathrm{\Delta }(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $\mathrm{\Delta }$-colorable if $G$ satisfies $(\mathrm{\mid }G\mathrm{\mid }+1)\mathrm{/}3\le \mathrm{\Delta }<\mathrm{\mid }G\mathrm{\mid }\mathrm{/}2$ and none of its components is a $K_{\mathrm{\Delta }+1}$.