Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to color the edges of G red or blue so that every vertex is adjacent to two red edges and two blue edges. Hint: Start by showing that G has a closed Eulerian walk.

2

Let G be a planar graph on 12 vertices with 24 edges in which all faces are bounded by cycles of length 3 or 4. How many triangles does G contain?

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.

Help me  with b and c

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

Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for each vertex v of G. (a) Prove that G must be a connected graph. (b) Prove that every vertex of G has degree at least 2.