Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for each vertex v of G. (a) Prove that G must be a connected graph. (b) Prove that every vertex of G has degree at least 2.

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: Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3.…

A: Step 1: Consider the provided question, (17) (i)

Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd length.

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Q: Let G be the (3,4)-complete bipartite graph. (a) List all non-isomorphic subgraphs with 5 vertices…

A: Part(a) Solution:Let be the (3,4) complete bipartite graph K3,4      These are the 11…

Q: Prove that every maximal matching in a graph G has at least a'(G)/2 edges.

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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: Suppose a graph G is regular of degree r, where r is odd. (a) prove that G has an even even number…

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Q: (d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even…

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Q: 4.1.3. (-) Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then…

A: Given: Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then G…

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: Let G be an undirected graph with n vertices. Let A(G) be the maximum degree of any vertex in G,…

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Q: .34. Consider the graph G in Fig. 8-58. Find: (a) degree of each vertex (and verify Theorem 8.1);

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Q: Prove that for every odd n ≥ 5 there exist a graph of n+1 vertices n of which have degree 3, and the…

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Q: 4. Prove: "The complement of the graph G is an Eulerian graph whenever G is also an Eulerian graph…

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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: Let G be a connected graph. Prove: (a) If G contains a cycle C which contains an edge e, then G – e…

A: Part (a):   A connected graph is a graph in which there is a path joining each pair of vertices, in…

Q: 2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a)…

A: The objective is to give an example of a graph with at leas four vertices, or prove that none…

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

A: Given: To prove the given statement as follows:

Q: Let G be a connected graph that has an Euler tour. Prove or disprove the following statements. If G…

A: Euler Circuit: A closed walk in a graph G that uses every edge exactly once is called an Euler…

Q: Find the smallest 3-regular graph G such that K(G) = 1.

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Q: True/False: If the graphs G1 and G2 are isomorphic, and G, is connected, then G2 must be connected…

A: If G1 and G2 are isomorphic , and G1 is connected

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

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Q: Prove that if G is a simple graph with n > 3 vertices and (",') +2 edges, then G is Hamiltonian. b)

A: In the given question we have to prove that a simple graph with vertices more than or equal to three…

Q: Prove or disprove: There exists some n > 1 such that there is a simple graph on n vertices, where…

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Q: Let G be a planar graph. Use Euler's Formula to prove that G must have a vertex of degree 5 or less.…

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Q: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Consider the general case, Let G be k-edge connected graph then we shall rove that G is k connected.…

Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

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Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

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Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if…

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Q: 3. (a) Is it possible to have a 4-regular graph with 15 vertices? If no, explain why. If yes,…

A: To find-  Is it possible to have a 4-regular graph with 15 vertices? If no, explain why. If yes,…

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

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Q: Let G be a 2-connected graph. Prove that if G has no even cycles, then G is isomorphic to an odd…

A: k-Connected graphs: A graph is said to be k-connected if it has more than k vertices and remains…

Q: Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible…

A: Please check the next step for details

Q: Draw a single graph G which satisfies all of the following properties. - All vertices of G are of…

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Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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Q: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

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Q: Let G be graph with 8 vertices and 10 edges. If every vertex of G has degree 2 or 3, then How many…

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Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: What is a(G) if G is a complete graph on n vertices?

A: It is given that G is a complete graph on n vertices. Also provided are some basic definitions about…

Q: prove that the degree of a regular complete tripartite graph kr,s,t with n vertices is given by 2n/3

A: Since the graph is tripartite with vertex division of r, s, t, we have three clusters of vertices…

Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…