Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to 2v-4, where v= number of vertices and f= faces of H. Show that for any integer v greater than or qual to 3, there exists a connected planar graph H that has v vertices and 2v-4 faces.

Q: Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7…

A: Given - G be a simple graph with 11 vertices, each of degree 5 or 6 To Prove - G has at least 7…

Q: Euler’s formula (v − e + f = 2) holds for all connected planar graphs. What if a graph is not…

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A: No, hypercube graph Q4 is not a planner graph.

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Q: Discrete Math-Graph Theory) Let G=(V,E₁∪E₂) be a simple graph such that G₁=(V,E₁) and G₂=(V,E₂) are…

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Q: Show that For n > 1 let G, be the simple graph with vertex set V(Gn) = {1,2,., n} in which two…

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Q: 4. (a) Define f : R×R→R×R by f(x,y) = (2x – y, x – 2y). Prove that f is one-to-one. Describe f-1.…

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A: (a)To find  the domain of the given function.

Q: consider ihe foriowing polýnomiai functions 2 f(x) =-2x"+8 .hx): 3x(x3ー4) 4 please graph BötH

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A: Let G be a connected graph which has 4 vertices of odd degree. The objective is to show that there…

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Q: 2n 4. Prove or disprove: for every n-vertex graph G, x(G) ≤ 2w(G) + a (G) (6 + 10.

A: There is n -vertex graph G. For the n vertex graph G, it has to be proven that χ(G) ≤2ω(G) + 2n/α(G)…

Q: Consider a bipartite graph on the set A = {1,2,3, 4, 5, 6} and B = {7,8,9, 10, 11, 12}, there is an…

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Q: Show that the average degree of a vertex in the triangulation is strictly less than 6.

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A: Consider the equation

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Q: The Euler's formula v - e + f = 2 holds for all connected planar graphs. What if the graph is not…

A: The formula for multiple connected component c is, v +f = e + c + 1

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: Complete a path straight at vertex a that ends at vertex c.

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A: Given a graph.

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Q: Use the formula deg(v) = 2|E(G)| to find the number of edges of the following VEV (G) graphs.…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three sub…