What does it mean to say that a graph $G$ is $k$-colourable? Define the chromatic number $\chi(G)$ of a graph $G$, and the chromatic number $\chi(S)$ of a closed surface $S$.

State the Euler-Poincaré formula relating the numbers of vertices, edges and faces in a drawing of a graph $G$ on a closed surface $S$ of Euler characteristic $E$. Show that if $E \leqslant 0$ then

$$\chi(S) \leqslant\left[\frac{7+\sqrt{49-24 E}}{2}\right\rfloor$$

Find, with justification, the chromatic number of the Klein bottle $N_{2}$. Show that if $G$ is a triangle-free graph which can be drawn on the Klein bottle then $\chi(G) \leqslant 4$.

[You may assume that the Klein bottle has Euler characteristic 0 , and that $K_{6}$ can be drawn on the Klein bottle but $K_{7}$ cannot. You may use Brooks's theorem.]