Let G be a 2-connected graph. Prove that if G has no even cycles, then G is isomorphic to an odd cycle C2k+1 for some k > 1.

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 (V, E) be a graph such that = • V ≥ 3, G has exactly 2 leaves, and in G all the non-leaf vertices have degree 3 or more. Prove that G has at least one cycle. You are not required to draw anything in your proof.

Can someone provide the proof for this? TIA

SOLVE STEP BY STEP AND IN DIGITAL FORMAT Prove that if G is a graph of order p and has size greater than (P-1) then G is connected. 2

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.

Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.