(a) Show that if $G$ is a planar graph then $\chi(G) \leqslant 5$. [You may assume Euler's formula, provided that you state it precisely.]

(b) (i) Prove that if $G$ is a triangle-free planar graph then $\chi(G) \leqslant 4$.

(ii) Prove that if $G$ is a planar graph of girth at least 6 then $\chi(G) \leqslant 3$.

(iii) Does there exist a constant $g$ such that, if $G$ is a planar graph of girth at least $g$, then $\chi(G) \leqslant 2 ?$ Justify your answer.