K6-Minors in Triangulations on the Nonorientable Surface of Genus 3

It is easy to characterize graphs with no Kk-minors for any integer k ≤ 4, as follows. For k = 1, 2, the problem must be trivial, and for k = 3, 4, those graphs are forests and series-parallel graphs (i.e., graphs obtained from K3 by a sequence of replacing a vertex of degree 2 with a pair of parallel edges, or its inverse operation.) Moreover, Wagner formulated a fundamental characterization of the graphs having no K5-minor [4]. However, a complete characterization of graphs having K6-minor seems to be a difficult problem, in general. In our talk, we consider the following problem: For a given triangulation G, which complete graph Kn is contained in G as a minor? It is easy to see that every triangulation G on any surface has a K4-minor, and that every triangulation G on any non-spherical surface has a K5-minor. The following are complete characterizations of triangulations on the projective plane and the torus with no K6-minor: