10. Let G = (V, E) be a loop-free connected undirected graph where V = {v1, v2, v3, . . . , vn},n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2 ≤ i ≤ n. Prove that G must have a cycle.
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10. Let G = (V, E) be a loop-free connected undirected graph where V = {v1, v2, v3, . . . , vn},
n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2 ≤ i ≤ n. Prove that G must have a cycle.
Step by step
Solved in 2 steps
- 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.8. Answer these two questions: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.
- 10. Type the answer correctly and do not use ChatGPT. Let G = (V, E) be a loop-free connected undirected graph where V = {v1, v2, v3, . . . , vn},n ≥ 2, deg(v1) = 1, and deg(vi) ≥ 2 for 2 ≤ i ≤ n. Prove that G must have a cycle.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.Q4. Prove that a complete bipartite graph Km,n is Hamiltonian if and only if m = n and m, n ≥ 2.
- please urgntly3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.9. The connected undirected graph G = (V, E) has 30 edges. What is the maximum value that|V | can have?
- 1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.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.2) The graph of f(x) = 0.5x(x − 4) (x - 7) is given below. fronte 10+ 10+ a) Draw on the graph the local linearization to f at x = 3 and label this function L(x).