State a result of Euler concerning the number of vertices, edges and faces of a connected plane graph. Deduce that if $G$ is a planar graph then $\delta(G) \leqslant 5$. Show that if $G$ is a planar graph then $\chi(G) \leqslant 5$.

Are the following statements true or false? Justify your answers.

[You may quote standard facts about planar and non-planar graphs, provided that they are clearly stated.]

(i) If $G$ is a graph with $\chi(G) \leqslant 4$ then $G$ is planar.

(ii) If $G$ is a connected graph with average degree at most $2.01$ then $G$ is planar.

(iii) If $G$ is a connected graph with average degree at most 2 then $G$ is planar.