# Does there exist 2-planar graph with chromatic number 8 or 9 or 10

A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $$|E(G)|\le 5(|V(G)|-2)$$. This implies that every 2-planar graph contains a vertex of degree at most 9 and thus every 2-planar is 10-colorable. On the other hand, $$K_7$$ is a 2-planar graph. So my question is whether there is a 2-planar graph with chromatic number 8 or 9 or 10? Thanks for pointers!

Note the $$K_8$$ is not 2-planar!

• Dmitry Karpov announced the proof that $\chi(G)\leqslant 9$, but this is not published yet. You may contact him directly: dvk0 at yandex dot ru Nov 22, 2019 at 10:54
• An earlier discussion of coloring $k$-planar graphs (but not helpful in answering this question) is mathoverflow.net/questions/168440/… Nov 22, 2019 at 11:46