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- 4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the k- regular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k – 1)/2 nearest vertices in each direction. Prove that < (G) = k. (Harary (1962a])1.2.11. (−) Prove or disprove: If G is an Eulerian graph with edges e, f that share vertex, then G has an Eulerian circuit in which e, f appear consecutively. a1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchang- ing the two partite sets) if and only if it is connected.
- 1.2.10. (-) Prove or disprove: a) Every Eulerian bipartite graph has an even number of edges. b) Every Eulerian simple graph with an even number of vertices has an even num- ber of edges.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.Each of the following assertions is either True or False. Write your answer legibly beside each question. Do not justify your answer.1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. עSEE MORE QUESTIONS
