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

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

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Q: Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e…

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

A: Let's assume that G is bipartite first. Odd cycles cannot exist in G because every subgraph of G is…

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: 3. A simple graph H is the square of a simple graph G if V (H) = V(G) and xy is an edge in H if and…

A: The simple graph H is the square of another simple graph G if V(H)=V(G). xy is an edge in H if and…

Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

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Q: (4) Show that it is not possible to define a graph with 5 vertices of degree 3, and 2 vertices of…

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Q: If we have two different non-crossing drawings of the same connected graph, is it possible for the…

A: The given question is related with planar graphs. Suppose we have two different non-crossing…

Q: Given a graph G = (V,E) its complement G is defined to be (V,E) where E = V{2} \ E. That is: a graph…

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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 "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: Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are…

A: note :Since you have posted multiple questions, we will provide the solution only to the first…

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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Q: Which of the following statements are true. Do not show your explanations. a) A tree is a graph…

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Q: 1. Does the graph below admit an Eulerian trail. If so, find one. If not, prove it.

A: You have asked multiple questions. So, I solve the first question for you. For the remaining part…

Q: Consider a connected graph G and let e = uv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂…

A: Let's go through each part of the problem step by step:Notation:- G: Connected graph.- e=uv∈E(G): An…