  A 1-plane graph is a graph embedded in the plane such that each edge is
crossed at most once. A 1-plane graph is optimal if it has maximum edge
density. A red-blue edge coloring of an optimal 1-plane graph $G$ partitions
the edge set of $G$ into blue edges and red edges such that no two blue edges
cross each other and no two red edges cross each other. We prove the following:
$(i)$ Every optimal 1-plane graph has a red-blue edge coloring such that the
blue subgraph is maximal planar while the red subgraph has vertex degree at
most four; this bound on the vertex degree is worst-case optimal. $(ii)$ A
red-blue edge coloring may not always induce a red forest of bounded vertex
degree. Applications of these results to graph augmentation and graph drawing
are also discussed.
