5. Let k be a fixed positive integer and let G = (V,E) be a loop-free undirected graph, where deg(v) ≥ k for all vЄ V. Prove that G contains a path of length k.
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- 10. Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and|V | = n, what is |E|?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.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.
- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.Please solve as much as you can7. a) How many different paths of length 2 are there in the undirected graph G in Fig. 11.43? b) Let G = (V, E) be a loop-free undirected graph, where V= {U, v2. ..., v) and deg(v,) = d,, for all I sisn. How many different paths of length 2 are there in G? Figure 11.43
- 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.Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1
- 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. 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.Explain thoroughly!!
