  A $1$-plane graph is a graph embedded in the plane such that each edge is
crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two
pairs of crossing edges share at most one end-vertex. An edge partition of a
$1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and
blue, such that both the graph induced by the red edges and the graph induced
by the blue edges are plane graphs. We prove the following: $(i)$ Every
NIC-plane graph admits an edge partition such that the red graph has maximum
vertex degree three; this bound on the vertex degree is worst-case optimal.
$(ii)$ Deciding whether a $1$-plane graph admits an edge partition such that
the red graph has maximum vertex degree two is NP-complete. $(iii)$ Deciding
whether a $1$-plane graph admits an edge partition such that the red graph has
maximum vertex degree one, and computing one in the positive case, can be done
in quadratic time. Applications of these results to graph drawing are also
discussed.
