2. Find a counterexample to the converse of Theorem 11.8 and show the full answer. Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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2. Find a counterexample to the converse of Theorem 11.8 and show the full answer.
Theorem 11.8: Let G = (V, E) be a loop-free graph with |V | = n ≥ 2. If deg(x) +
deg(y) ≥ n − 1 for all x, y ∈ V, x̸ = y, then G has a Hamilton path.
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- 8. Let G be the graph below. (а) Does G has the Hamiltonian path? (b) Does G has the Hamiltonian cycle?8.2-1b)2. Answer the following questions. a. If a graph is Hamiltonian, is it necessarily Eulerian as well? If yes, explain why. If no, provide a counterexample. b. If a graph is Eulerian, is it necessarily Hamiltonian as well? If yes, explain why. If no, provide a counterexample.
- 4) Use the superposition principal to draw a graph of each. a) y = f(x)+ g(x) b) y = g(x)-f(x) f(x) -4 g(x) f(x) YA 4 2 0 2 4 y 4 2 -4 0 A 2 g(x) -4 2 2 4x 4 xKk.212.1. Consider the following graphs. G₁ G₁ G3 (a) Which of the previous graphs have a Hamiltonian path? (b) Which of the previous graphs have a Hamiltonian cycle? (c) Which of the previous graphs have an Eulerian trail? (d) Which of the previous graphs have an Eulerian circuit? G₂ #
- 1. What are the eventually fixed points for A(x) = [x? 2. Let F(x) = 1-x. Show that 0 lies on a 2-cycle for this function. 3. Consider the function F(x) = /x-2/. a. What are the fixed points for F? b. If m is an odd integer, what can you say about the orbit of m? c. What happens to the orbit if m is even? 4. How many fixed points does the function T(x) = tan(x) have? 5. Does the function F(x) = -x have a cycle of prime period 2? 6. For which values of o and b does the linear function L(x) = ax + b have a cycle of prime period 2? 7. let T(x) be Tent Map then find (a)Find a formula for T(x). (b). Sketch the graphs of T and T. (c) Find all fixed points for Tand T. (d). Find an explicit formula for T(x) and sketch the graph of T. (f). What does the graph of T look like?6.38) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.Please may I have the correct solution for part b of this question.