Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.

Q: Let G = (V, E) be a planar graph such that |V |< 19. Show that if the length of the shortest cycle…

A: For a given graph G=V,E where V≤19 that shortest cycle in G is 5, then we need to prove that G has a…

Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

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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 the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are

A: Among the generalized Petersen graphs, having n prism G p,1 on the Durer graph G 6.2 the…

Q: Let G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and the other 7…

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Q: Let H be a graph If H has at least one vertex of degree greater than zero then there are two of even…

A: Given a graph representing H. Statement: If H has at least one vertex of degree greater than zero…

Q: 4. Prove: "Show that the number of edges in a simple planar graph of order n is at most 3n – 6".…

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Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

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Q: Let x and y be two adjacent vertices in the complete bipartite graph Kn,n,     n ≥ 3. Find the…

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Q: Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane…

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Q: Give an example of a graph with five vertices of degree 1, and six vertices of degree 2. If you…

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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: Draw a simple graph (or argue why one cannot exist) that (a) has 6 vertices, 12 edges, and is…

A: (a)Such a graph is not possible as we can seeK3 has 3 vertices 3 edgesK2 has 2 vertices and 1 edgeK4…

Q: Give an example of a graph G of order 6 and size 10 such that the minimum and maximum degree of a…

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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

Q: Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

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Q: a) Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex…

A: Part(a): G is a simple graph with n vertices and degree of each vertex in G is at least n-12. We…

Q: Draw a graph G with vertex and edge set  , where the endpoints of e1 are v1 and v2, the endpoints of…

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Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

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

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

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: 1. show that every simple finite graph has two vertices of the same degree. 2.prove that the degree…

A: Disclaimer:  Since you have asked multiple question, we will solve the first question for you. If…

Q: Prove that any graph with at least two vertices must have two vertices of the same degree.

A: Let G be a graph with n vertices where n≥2 To prove: G must have two vertices of same degree  On the…

Q: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

A: We have to prove :       If u is a vertex of odd degree in a graph, then there exists a path from…

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: Prove the following and make a sketch. a) Prove that every Eulerian graph of odd order has three…

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Q: 2. Prove the following and make a sketch. a) Prove that every Eulerian graph of odd order has three…

A: 1. Since G is connected, every vertex has positive degree. Since G has an Eulerian cycle, every…

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

A: Consider the equation

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: For any connected graph G of order 55, if x is a vertex of G, then the number of components of G-x…

A: A connected graph is graph that is connected in the sense of a topological space, i.e., there is a…

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: 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: If G has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is the problem of graph theory.

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: 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: Let G be a graph with exactly 10 vertices and 27 edges. Suppose that each vertex has degree 3, 5, or…

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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 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: 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: Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).

A: Since you have asked multiple questions, we will solve the first one for you. If you want any…

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: Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for…

A: If we are confident with our definition we can easily attempt like these questions, G is connected…

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…