Past exam of the geometry course of the University of Oxford / 2011 / 1 / c / ii

The simplicial intersection conditions prohibit loops and multiple edges between the same pair of vertices, so the one-skeleton is a simple graph andUsing the preceding identity gives $V^2-7V+6\chi \ge 0$. A closed surface has $\chi \le 2$ and a simplicial triangulation has $V\ge 4$. The smaller root is at most three, so this inequality forces $V$ to lie above the larger root:For completeness, fewer than four vertices could give at most one triangle under the intersection rules, which cannot form a closed surface.