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. Answer these two questions: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.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.
- 4. Prove that for each n€ Z+ there exists a loop-free connected undirected graph G = (V,E), where |V| 2n and which has two vertices of degree i for every 1≤i≤n. =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.)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.
- 4) a) Two bugs follow the paths ri(t) = and 20t-10 r2(t) =. Determine if their paths cross. If so, at what point? t+23. [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.Please help with questions b & c
- a) Determines the number of Zeropoints5.1.20. () Let G be a graph whose odd cycles are pairwise intersecting, meaning that every two odd cycles in G'have a common vertex. Prove that x(G) <5.Prove that every graph G with n vertices and chromatic number k = x(G) has at most · (n² – ) | edges. (Hint: What is the maximum number of edges possible? How many edges must be missing?) (Hint: You can use as an axiom thatE=1n, where E=1n; = n is minimized when each n¡ = n/k.)