  An odd coloring of a graph $G$ is a proper coloring such that any
non-isolated vertex in $G$ has a coloring appears odd times on its neighbors.
The odd chromatic number, denoted by $\chi_o(G)$, is the minimum number of
colors that admits an odd coloring of $G$. Petru\v{s}evski and \v{S}krekovski
in 2021 introduced this notion and proved that if $G$ is planar, then
$\chi_o(G)\le9$ and conjectured that $\chi_o(G)\le5$. More recently, Petr and
Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the
plane so that each edge is crossed by at most one other edge. Cranston,
Lafferty and Song showed that every $1$-planar graph is odd $23$-colorable. In
this paper, we improved this result and showed that every $1$-planar graph is
odd $13$-colorable.
