b please The statement of Ore's Theorem is:If G is a graph of order n ≥ 3 such that for all pairs of distinct non-adjacentvertices x and y, deg x + degy ≥ n, then G is Hamiltonian.The full strength of the proof of Ore's theorem was not needed to prove this result.Use either the statement or arguments from the proof to prove the following:(a) If G is a graph of order n ≥ 3 such that 8(G) ≥ n/2 then G is Hamiltonian.(b) Let x and y be distinct nonadjacent vertices of a graph G of order n such thatdeg x + deg y ≥ n. Then G+ xy is Hamiltonian iff G is Hamiltonian.

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Answered: The statement of Ore's Theorem is: If G…

Linear Algebra: A Modern Introduction

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Chapter6: Vector Spaces

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3. [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.
Prove or disprove: There exist two connected graphs G and H both of order at least 3 and neither of which is Eulerian such that G + H is Eulerian.
If G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v # w such that deg(v) = deg(w). Present a counterexample, if G is allowed to have self-loops.
Q4. Prove that a complete bipartite graph Km,n is Hamiltonian if and only if m = n and m, n ≥ 2.
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.
9. Which of the following statements are true? T(G) is connected for every connected graph G. If T (G) has more than one vertex then it has a cycle. If G and G'are isomorphic then so are T (G) and T(G'). □ IfT(G) and T (G') are isomorphic then G and G' are isomorphic
Let G be a bipartite Prove that a(G) = graph of order n. if and only if G has a perfect matching.
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
Let G (V, E) be a graph such that = • V ≥ 3, G has exactly 2 leaves, and in G all the non-leaf vertices have degree 3 or more. Prove that G has at least one cycle. You are not required to draw anything in your proof.
Consider the given graphs: U18 u₂ 16 15 ●113 14 Vg V7 V₁ V6 V/₂ V5 V3 V4 G H Click and drag the statements in correct order either to show that these graphs are not isomorphic or to find an isomorphism between them using paths. Proceeding along similar lines, complete the bijection with us → V8, U6 → V7, and U7 → VS. Having thus been led to the only possible isomorphism, check that the 12 edges of G exactly correspond to the 12 edges of H, and the two graphs are isomorphic. Now u, is adjacent to uz and us, and v₂ is adjacent to v₁ and vi, so u₂ v. and us > V Having thus been led to the only possible isomorphism, check that the 12 edges of G do not correspond to the 12 edges of H; hence the two graphs are not isomorphic. Since u4 is the other vertex adjacent to us and v4 is the other vertex adjacent to V3, U4 → V4. So try to construct an isomorphism that matches them, say u₁ → V₂ and Us → V6.
). 4. Rectangle DEFG with vertices D(-4, 3), E(O, 2), F(-2, -6), and G(-6, -5): (x, y) (x+ 4, y + 1) D'( E'( . F' G' - onent form. 7. OGina Wison (AI Things Algebra, LLC). 2015-2018
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.)

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