How can I prove the following: Let G be a 2-connected graph. If e and f are parallel edges in G, then G\e is 2-connected. (Without deleting edges or vertices. However edge contraction is allowed)

prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

6. Determine whether the following pair of graphs is isomorphic. Exhibit an isomorphism ( show the mapping between vertices) or provide an argument that none exists. V1 V3 V5 V1 V2 V5 VA V3 V4 V2 G1 G2

For which of the following does there exist a simple graph G = - (V, E) satisfying the specified conditions? Select one: O A. It has 7 vertices, 10 edges, and more than two components. B. It has 8 vertices, 8 edges, and no cycles. O C. It has 6 vertices, 11 edges, and more than one component. O D. It is connected and has 10 edges, 5 vertices and fewer than 6 cycles. O E. It has 3 components, 20 vertices and 16 edges.

Give a path of maximum length in G.

True or False: There exists a simple graph with 9 vertices each of degree 5.

4. Determine if the following graphs are isomorphic or not? If yes, show how are they isomorphic. If not also explain your justification. (hint: You can label your vertices)

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 and let e1 and e2 be edges.  Show that if deleting e1 and e2 disconnects vertices u and v, then any cycle containing both u and v must contain both e1 and e2.  One approach:  You could apply to the graphs G-e1 and G-e2 the fact that if e is a bridge whose removal disconnects u and v, then any path connecting u and v must contain e.

Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.