State Euler's formula relating the number of vertices, edges and faces in a drawing of a connected planar graph. Deduce that every planar graph has chromatic number at most $5 .$

Show also that any triangle-free planar graph has chromatic number at most 4 .

Suppose $G$ is a planar graph which is minimal 5 -chromatic; that is to say, $\chi(G)=5$ but if $H$ is a subgraph of $G$ with $H \neq G$ then $\chi(H)<5$. Prove that $\delta(G) \geqslant 5$. Does this remain true if we drop the assumption that $G$ is planar? Justify your answer.

[The Four Colour Theorem may not be assumed.]