Highlight a Hamiltonian cycle, if it exists, in each of the following graphs, or state that no Hamiltonian cycle exists (no explanation is needed in this case). (a) (b)
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- 31. In Example 2.35, describe all possible configurations of lights that can be obtained if we start with all the lights off.What does Dirac’s Theorem state? Explain how it guarantees that the graph is HamiltonianUsing the following graph answer the following; (a) Find the smallest number of colors required to color it. (b) If possible, find a Hamiltonian cycle. (c) Show that the sum of the degrees of the vertices is twice as large as the number of edges.
- Theorem; If n ≥ 3 and if d(xk) ≤ k < ½*n → d(xn-k) ≥ n - k Then graph G = (X,E) contains a Hamiltonian cycle. Proof please !(5) (a) Let a and b be two elements of Sn with the same cycle structure. Explain how to produce an element g of Sn which conjugates a to (b) Show that even if a and b belong to An, we cannot always take g € An.The picture on the left is that of an icosahedron, a solid object whose faces consist of 20 congruent equilateral triangles. By stretching the base triangle and flattening, the icosahedron determines a graph in the plane (as shown on the right side of the figure). Find a Hamiltonian cycle in this graph.
- Consider the permutation (taqgblvxh)(n skw fue)(ycodjr)(pz mi). (a) Showing your working, and working entirely in disjoint cycle notation, give the disjoint cycle notations for T and for T-1, and then compute T*(x) and -1(c). (b) Justifying your answer, find two positive powers a and B such that n° = T = T (c) Explain how a monoalphabetic substitution cipher works. Illustrate your explana- tion by using the permutation 7 above to encrypt the plaintext It is not that secrets are never needed in security, but they are never desirable.(b) State Heine-Borel Theorem.Conjugate elements. Let 0,T E Sn. Define d = TOT. (o and d are called conjugate elements; see Chapter 6.) (a) Show that if o (i) = j, then 8(r (i)) = T(j). (b) Explain that the previous part says that if you apply T to each of the entries in the cycle notation of o, then you get the cycle notation for 8. In other words, if o has cycle decomposition (a1 a2 aki k1)(bị b2 bk2) · ·· . .. ... 3. The Alternating Groups Then d has cycle decomposition (7(a1) T(a2) ·… (c) Illustrate the previous part by letting o = (1 6 3)(7 5 2), and quickly writing down the cycle decomposition for TOT. Check your answer by actually finding the product rOT. T(bk2)) .... (1 4 3 8)(2 6 5), т — T(ak,))(t(b1) T(b2) .. ..