Tightness of Kainen’s lower bound at minimal triangulations

Establish that Kainen’s lower bound on the surface crossing number is tight for the complete graph K_{M(g)} drawn on the orientable surface S_g, where M(g) = ceil((7 + sqrt(1+48g))/2) is the minimum number of vertices in a triangulation of S_g; equivalently, construct a minimal triangulation of S_g on M(g) vertices such that each missing edge of K_{M(g)} can be added using exactly one crossing.

Background

Jungerman and Ringel constructed minimal triangulations of orientable surfaces S_g (g ≠ 2) and identified the minimum number of vertices M(g) needed for a triangulation. Kainen’s lower bound provides a general lower bound on the surface crossing number cr_g(G) in terms of |V|, |E|, and g.

The paper references a conjecture previously posed by the author asserting that for the complete graph on M(g) vertices, Kainen’s lower bound is tight, meaning there exists a minimal triangulation of S_g in which every missing edge of K_{M(g)} can be added with only one crossing. Theorem 3 in the paper verifies this tightness for some surfaces, leaving the general case open.