  The Chen-Lih-Wu Conjecture states that each connected graph with maximum
degree $\Delta\geq 3$ that is not the complete graph $K_{\Delta+1}$ or the
complete bipartite graph $K_{\Delta,\Delta}$ admits an equitable coloring with
$\Delta$ colors. For planar graphs, the conjecture has been confirmed for
$\Delta\geq 13$ by Yap and Zhang and for $9\leq \Delta\leq 12$ by Nakprasit. In
this paper, we present a proof that confirms the conjecture for graphs
embeddable into a surface with non-negative Euler characteristic with maximum
degree $\Delta\geq 9$ and for planar graphs with maximum degree $\Delta\geq 8$.
