2. Find a counterexample to the converse of Theorem 11.8. 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.
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. Answer these two questions:7. [10 marks] Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G of length 5. We show how to find a longer cycle in G. (a) Let x be a vertex of G that is not on C. Show that there are three C-paths Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect C at their final vertices. (b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C. (c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.7. [10 marks] Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G of length 5. We show how to find a longer cycle in G. Ꮖ (a) Let x be a vertex of G that is not on C. Show that there are three C-paths Po, P1, P2 that are disjoint except at the shared initial vertex x and only intersect C at their final vertices. (b) Show that at least two of Po, P1, P2 have final vertices that are adjacent along C.
- 3. [10 marks] Let Go = (V,E) and G₁ = (V,E₁) be two graphs on the same set of vertices. Let (V, EU E1), so that (u, v) is an edge of H if and only if (u, v) is an edge of Go or of G1 (or of both). H = (a) Show that if Go and G₁ are both Eulerian and En E₁ = Ø (i.e., Go and G₁ have no edges in common), then H is also Eulerian. (b) Give an example where Go and G₁ are both Eulerian, but H is not Eulerian.7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)8. If the graph of h(x) goes through the points A(-16,4), B(-4,0), C(0,-4) and D(4,16), then how many of the following statements are true for the graph of h(x) ? Point A will be mapped onto A'(-16,2). II. Point B will be an invariant point. III. Point C will be an invariant point. IV. Point D will be mapped onto D' (2,16) I. А. one B. two C. three D. four ZEBRA Mild Ink M IL DLI NER. -V5 А. V5 В. 5 2/5 С. - 2/5 D. Tiw doidw 10. If cos0 = V3 sin O then the exact value of tan 20 is: A. -V3 В. 2/3 D. 3 С.
- 1. Show that y₁= ez and y2 = ze form a linearly independent set on the interval (-00,00).3. Find ƒ(2), fƒ(3), ƒ(4), and f(5) if f is defined recur- sively by f(0) = -1, f(1) = 2, and for n = 1, 2, ... a) f(n+1) = f(n) +3f(n-1). b) f(n + 1) = f(n)² ƒ (n − 1). c) f(n+1) = - 3ƒ (n)² – 4ƒ (n − 1)². d) f(n + 1) = f(n − 1)/ƒ (n).10. Given h(x)(x 1)3 (x1)4 (x 3)2 (a) Determine the degree and end behavior. As χ →_oo : yos, t ve ).f.m.t (As χ→ oo : posit-ve ) hf7xit Degree: (b) Find the x-intercepts, the multiplicity of each root, and state whether the graph bounces of the x -axis or crosses through the x -axis. Los 3 molthiylietes The grap ounces of thex-axis
- 5.- State A for the linear transformation T(ü) = Aŭ " (1) - |-| -c+d P 1 -1 a) A = 1 b) A = 1 -1 1 1 c) A = |-1 1 d) None of the above.x 6. Copy each graph of f(x) and g(x). Then, apply the superposition principle to graph f(x) + glx) Give the domain and range of f(x) + g(x), a) y fx) g(x) 0. n. b) flx) 4 g(x) 20 2 4 6. -4 2. 6. 2. 4. 2. 2, 6.Let G = (V, E) be a connected, undirected graph. Let A = V, B = {1,..., n − 1}, and ƒ(v) = deg(v). Select all that are true. a) f : A → B is not a function b) f: A → B is a function but we cannot always apply the Pigeonhole Principle with this A, B c) f: AB is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above