A graph is 1-planar if it can be drawn on the plane such that each edge is crossed at most once.

Let $G$ be a 1-planar bipartite graph with $n~(n > 4)$ vertices and $m$ edges. Karpov 1 showed that $m \ge 3n − 8$ holds for even $n \ge 8$ and $m \ge 3n − 9$ holds for odd $n \ge 7$.

• 1. D. V. Karpov. Upper bound on the number of edges of an almost planar bipartite graph. J. Math. Sci., 196:737–746, 2014.

So the minimum degree of any 1-planar bipartite graph is at most $5$. Here is my question.

• Construct a 5-regular bipartite 1-planar graph.

I note that $5n\le2(3n-8)$ impling that $n\ge 16$. Maybe we will find such graph with $16$ vertices.

The problem comes from planar bipartite graphs. Any planar bipartite graph has minimum degree 3. The smallest order of 3-regular planar bipartite graph is 8; see the below graph.