The density of a tripartite 1-planar graph

1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $$n$$-vertex 1-planar graph is at most $$4n-8$$, and the maximum number of edges of a bipartite $$n$$-vertex 1-planar graph is at most $$3n-8$$. I remember that there is also a result on the the maximum number of edges of a tripartite $$n$$-vertex 1-planar graph but I can't find the source now. Does anyone know this result or the source? Thanks in advance!

• By "density", do you mean "number of edges"? Jan 19, 2021 at 6:26
• Yes, it is the number of edges. Jan 19, 2021 at 8:49