Prove that the following conditions on a connected graph Γ are equivalent. Γ  is a tree. Given any two vertices v and w in Γ, there is a unique reduced edge path from v to w. For every edge e # E(Γ), removing e from Γ disconnects the graph. (Note: Removing e does not remove its associated vertices.)

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Q: 7. Suppose a graph G = (V, E) has n vertices and m edges (i.e., |V| = n and |E| = m). Further…

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A: Given, Connecting two nodes u and v in a graph G by a new edge creates a new cycle such that if and…

Q: Use Ore's theorem to prove that any n-vertex graph G for which |E(G)|> (",') +2 is Hamiltonian.

A: answer is in next step

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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: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A: Let v0 and v1 be distinct vertices of a graph G=V,E. Assume that there exists path in G from v0 to…

Q: simple path in G from v0 to v1

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Q: b) Is the following graph isomorphic to Q4? If yes, give an isomorphism from Q4 to this graph.…

A: 2b ) Given graph is Here number of vertices in the given graph is 12 and number of vertices in Q4…

Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: Q10. Using Corollary 3.1.24 of the book, to prove that a graph G is bipartite if and only if a(H) =…

A: Given: α(H)=β'(H) To find: Isolated vertices

Q: a. Let V = (x, y, z, u, v). How many di exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ =…

A: (a) We are given that the set of vertices is V=x, y, z, u, v. We are asked to find the number of…

Q: Prove or disprove the following. Let G = (V, E) be a simple undirected graph without loops, where…

A: Definition Degree of vertex: The degree of a vertex is the number of edges connecting it.…

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: The complement of a graph G is a new graph formed by removing all the edges of G and replacing them…

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Q: Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where DI

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Q: 4.1.14. (!) Let G be a connected graph in which for every edge e, there are cycles C1 and C2…

A: The given question is related with graph theory. Given that G is a connected graph in which for…

Q: 4.1.5. (-) Let G be a connected graph with at least three vertices. Form G' from G by adding an edge…

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Q: Prove or disprove the following. Let G = (V, E) be a simple undirected graph without loops, where…

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Q: Consider the following statements: S: There exists a connected simple graph with 6 vertices and 5…

A: We have to verify the given statements.

Q: Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Diracs…

A: Dirac's theorem Let G = (V,E) be a graph with n vertices in which each vertex has degree at least…

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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: Let T = (V, E) be a tree with V > 3 and k > 2 teaves. Prove that we can add k – 1 edges to T to…

A: Let TV, E be a tree with V≥3 and k≥2 leaves. Show that k-1 edges can be added to the graph of T to…

Q: Prove that every simple graph G= (V, E) with |V] > 2 has at least two vertices of equal degree. %3D…

A: We have a simple graph, with V≥2, that is, the number of vertices is greater than or equal to 2.…

Q: (5) Show that we cannot build a bipartite graph with bipartition V(G) = Vị UV½ such that |V = 3 and…

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Q: Prove if n greater than or equal 2 and G is connected with 2n odd vertices then G has n open walks…

A: Hamiltonian Path: If we remove any one edge from a Hamiltonian circuit. then we left with a path.…

Q: Prove that if G = (V, E) is a connected graph such that |E |V| – 1, then G is a tree. %3D

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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: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

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Q: Prove that the following graph μ is bipartite. w V u fl a b с

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Q: For the given graph G, which of the following statements is false? A B) (F a) G is a simple directed…

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Q: 4.1.17. Prove that deleting an edge cut of size 3 in the Petersen graph isolates a vertex.

A: Edge Connectivity: λ(G) or κ′(G) is the minimum size of edge set F s.t. G\F has more than one…

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

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Q: a. Let V = (x, y, z, u, v). How many distinct graphs are there on the exactly 6 edges? (Note that…

A: (a) The set of vertices given is as follows: V=x, y, z, u, v We know that the total sum of degrees…

Q: Show that if an edge e is in a closed, trail of G, then e is in a cycle of G

A: Use the following definitions, to obtain the solution. A trail is a sequence of vertices and edges…

Q: Discrete math graph theory

A: Two graphs S and T are isomorphic if there exists a bijection f between the vertex sets of S and T…

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: Prove that if there

A: To prove that if there exists at most one path between any two vertices of a simple graphG, then G…

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: 1. Let C(G,k) denote the decision problem of whether the undirected graph G = subset of vertices V'…

A: Let C(G,k) denotes the decision problem, Let G is the given graph and k be the subset of the graph…

Q: a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6…

A: We have given graph V{x,y,w,u,z} We need to calculate a)graph having 6 edges. b) The trail which is…

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…