Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and edge e of G, v and e lie on a common cycle of G.

Q: Let G be a connected graph. Prove that G is Eulerian if and only if each of the blocks of G is…

A: The objective of the question is to prove that a connected graph G is Eulerian if and only if each…

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: Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G) \ {e}. Then e is…

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Q: Prove that every connected graph with m vertices has at least (m-1) edges, for every m>1

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Q: Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial…

A: Theorem:(Statement) For every graph G, Proof: Let  us consider two  cases.Case1: G is a  trivial…

Q: Euler cycle.

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

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: complement

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Q: Let G be a simple connected graph.  (a) Suppose that G has t blocks. Prove that there is an ordering…

A: First, we need to understand the problem. A block in a graph is a maximal connected subgraph without…

Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

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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 for every even integer n ≥ 2 that there exists a graph G of order n such that G has n/2 pairs…

A: According to the question, In graph theory, two vertices and u are similar if there is an…

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: Let G be a graph with n ≥ 3 vertices that has a clique of size n − 2 but no cliques of size n − 1.…

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Q: "Let G be a simple graph. The graph G has no induced subgraph isomorphic to P3 if and only if every…

A: The given statement is: "Let G be a simple graph. The graph G has no induced subgraph isomorphic to…

Q: Let n be a large positive integer. Choose the best option to finish the next sentence. n Σ The sum…

A: Please find the next step for detailed answer.

Q: Let G be a bipartite Prove that a(G) = graph of order n. if and only if G has a perfect matching.

A: Let G be a bipartite graph of order n. We need to prove that  if and only if G has a perfect…

Q: Let G be the graph on vertices {2, 3,..., n}, where a and b are joined by an edge iff gcd (a, b) =…

A: Step 1:Step 2:

Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: Prove: “The complement of the graph G is an Eulerian graph whenever G is also  an Eulerian graph of…

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