I need your help in the next problem:
I use $N_g$, $g \geq 1$, to denote the nonorientable surface which can be constructed by inserting $n$ cross-caps on the sphere (these cannot be embedded in $\mathbb{R}^3$).
I need to show that if a simple graph $G$ is embedded in the nonorientable surface $N_g$, $g \geq 1$, then:
$$\chi(G) \leq \frac{7 + \sqrt{1 + 24g}}{2},$$
where $\chi(G)$ denotes the chromatic number of $G$.
Thanks.