Optimal 1-planar graphs which triangulate other surfaces

A graph G is called a 1-planar graph if it can be drawn on the plane so that each edge includes at most one crossing. It is not so difficult to see that |E(G)| ≤ 4|V (G)| − 8 for any 1-planar graph. In particular, a 1-planar graph G is said to be optimal if the equality holds. Suzuki has already proved that there exists an optimal 1-planar graph which can be embedded on the orientable closed surface of genus g as a triangulation for any positive integer g; such a 1-planar graph must have exactly 2 + 6g vertices. On the other hand, he has shown that there exists no such graph for the non-orientable closed surfaces of genus 1, 2 and 3. Furthermore, we prove the non-existence of such a graph for the non-orientable closed surface of genus 4, carrying out a computer experiment to generate all possible triangular embeddings of optimal 1-planar graphs with 14 vertices.