  An \emph{equitable coloring} of a graph is a proper vertex coloring such that
the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu
conjectured that every connected graph $G$ with maximum degree $\Delta \geq 2$
has an equitable coloring with $\Delta$ colors, except when $G$ is a complete
graph or an odd cycle or $\Delta$ is odd and $G=K_{\Delta,\Delta}.$ Nakprasit
proved the conjecture holds for planar graphs with maximum degree at least 9.
Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph
with maximum degree at least 8 and for every planar graph without $C_4$ and
$C_5$ with maximum degree at least 7.
  In this paper, we prove that the conjecture holds for planar graphs in
various settings, especially for every $C_3$-free planar graph with maximum
degree at least 6 and for every planar graph without $C_4$ with maximum degree
at least 7, which improve or generalize results on equitable coloring by Zhu
and Bu. Moreover, we prove that the conjecture holds for every planar graph of
girth at least 6 with maximum degree at least 5.

Combinatorics