Prove that a graph is two-colorable (bipartite) if and only if it contains no odd-length cycle

16 Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <4 (Exercise 19 in Section 1.4) to prove that every planar graph with less than 12 vertices can be 4-colored.

Draw graph G and its complement, showing that at least one of G and it's complement, G', is connected.

I have to prove the following Corollary: "Let u and v be vertices of a 2-connected graph G. Then there is a cycle of G that contains both u and v.”   My idea of the proof is the following: Given vertices u and v, I want to show that there is a cycle containing both. Then by contradiction, I could assume that there exists u and v sucht that there are no cycles C including u and v.  However, I'm not quite sure that this is the correct approach. I would appreciate some help to prove this corollary. Thank youuu

Let G be a graph. Prove that if G has a closed walk of odd length, that the vertices and edges of the walk contain a cycle.

Let G be a connected graph of order n and size n. Prove that G contains a single cycle.