Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e is removed from G to create the graph G e, then there exists a path from v1 to v2 in G – e. If e is removed from G to create the graph G e, then there does not exist a path from v1 to v2 in G – e. O If e is removed from G to create the graph G - e, then it is not possible to determine if there exists a path from vị to v2 in G – e.

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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A: Let us prove above result

Q: 1. Below is a drawing of a graph G = (VG, EG), and then the vertex set and edge set of a graph H…

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Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Euler cycle.

A: AnswerSuppose we have some simple graph G, with odd n>, 1 nodes, and that this graphhas an Euler…

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: Suppose that G = (V, E) is a directed graph. A vertex w E V is called reachable from a vertex v € V…

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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: 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: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

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Q: Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and…

A: Proof:Sufficiency:Suppose that G is 2-connected and for any vertex v and edge e of G, v and e lie on…

Q: Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a…

A: First, we need to understand the problem. We are given two paths P1 and P2 of maximum length in a…

Q: A graph X is called self-complementary if it isomorphic to its complement. For a self-complementary…

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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…

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

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Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

Q: Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G'…

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Q: Show that every simple graph on n vertices is isomorphic to a subgraph of Kn.

A: We need to prove that every simple graph on n vertices is isomorphic to a subgraph of Kn (complete…

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Q: Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

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Q: Show that an edge e is a bridge of a graph G if and only if {e} is a bond of G.

A: We prove that an edge e is a bridge of a graph G if and only if {e} is a bond of G.  We need to…

Q: 5. Consider the undirected graph G = {V, E}, where V, the set of nodes, is {1, 2, 3, 4} and E, the…

A: 5. Given that the undirected graph G=(V,E) where V , the set of nodes, is {1, 2, 3, 4} and E, the…