(i) State a result of Euler, relating the number of vertices, edges and faces of a plane graph. Show that if $G$ is a plane graph then $\chi(G) \leq 5$.

(ii) Define the chromatic polynomial $p_{G}(t)$ of a graph $G$. Show that

$$p_{G}(t)=t^{n}-a_{1} t^{n-1}+a_{2} t^{n-2}+\ldots+(-1)^{n} a_{n}$$

where $a_{1}, \ldots, a_{n}$ are non-negative integers. Explain, with proof, how the chromatic polynomial is related to the number of vertices, edges and triangles in $G$. Show that if $C_{n}$ is a cycle of length $n \geq 3$, then

$$p_{C_{n}}(t)=(t-1)^{n}+(-1)^{n}(t-1) .$$