# TU Wien:Diskrete Mathematik für Informatik VU (Drmota)/Prüfung 2020-12-11

### Let S be a compact orientable surfaces with genus >= k. State the Euler characteristic of S. (2 Points)

${\displaystyle \chi (S)=2-2k}$

### Let G be a connected graph that is embedded on S. State the formula relating the number of vertices, faces, and edges of G. (2 Points)

${\displaystyle \alpha _{0}(G)-\alpha _{1}(G)+\alpha _{2}(G)=\chi (S)(\alpha _{0}=|V|,\alpha _{1}=|E|,\alpha _{2}=|F|)}$

### Suppose that G has n faces, and each face is bounded by exactly 4 edges. Calculate the number of vertices and edges of G. (5 Points)

1. edges = 2* #faces = 2n
2. vertices = #edges - #faces + \chi (S) = 2n-n + (2-2k) = n + 2-2k