1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).
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- 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.1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. ע5.1.20. () 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.
- 1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchang- ing the two partite sets) if and only if it is connected.1.2.18. (!) 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.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.
- ). 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). 2015-20184) Let a = (-3, 0, -4) and 6 = (-3, 3, 1). Find the projection of b onto a.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. a
- 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.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.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.
