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.”

Q: Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7…

A: Given - G be a simple graph with 11 vertices, each of degree 5 or 6 To Prove - G has at least 7…

Q: G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v w ch that…

A: Graph without self loop is called simple

Q: If G1 and G2 are isomorphic, must they have the same number of vertices? If G1 and G2 have the same…

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Q: Answer the following questions: a.Suppose that a connected bipartite planar simple graph has e…

A: "Hey ,since there are multiple questions posted, we will answer first question. If you want any…

Q: 4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does…

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Q: 33. Which of the following statements is/ are true? I. Adding an edge to a path graph will always…

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Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

Q: The most famous theorem in graph theorem is the following: Theorem 1. The vertices of a planar graph…

A: No, it is not true that all graphs with χG<4 are planar. it is known that Peterson graph is not…

Q: "If G has a walk that uses all the edges, then G is connected." I know that this statement is false…

A: If G has a walk that uses all the edges, then G is connected Clearly this statement is false For…

Q: Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: How can I prove the following: Let G be a 2-connected graph. If e and f are parallel edges in G,…

A: To prove that G\e is 2-connected, we need to show that for any two vertices u and v in G\e, there…

Q: How can I prove "A graph is k-colourable if and only if it is k-partite.”?

A: As per the question we have to prove that a graph is k-colourable if and only if it is k-partite. To…

Q: can you please answer c and d

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Q: 3. (Exercise 3.4) Prove Lemma 3.3: "Let G be a 2-connected graph. If e and f are parallel edges in…

A: Let G be a 2-connected graph. e and f are parallel edges in G. Let H= G\e We have to prove that if…

Q: Problem 10.2. Can a graph with 6 vertices have vertices with the following degrees: 3, 4, 1, 5, 4,…

A: Kindly refer to Step 2 for the solution.

Q: The Euler's formula v - e + f = 2 holds for all connected planar graphs. What if the graph is not…

A: The formula for multiple connected component c is, v +f = e + c + 1

Q: Prove that cycles are bipartite for an even number of vertexes. But they are not bipartite for the…

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