1 2. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 125 Order all of the following sentences in the most nat

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 125 Order All Of The Following Sentences In The Most Nat 1 (63.15 KiB) Viewed 73 times

1 2. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 125 Order all of the following sentences in the most natual way so that they form a proof for the following statment: The above graph does not have a Hamiltonian cycle. Choose from these sentences: Your Proot To find a contradiction, observe that this graph has the property that every edge is between one odd vertex and one even vertex. This is a contradiction to the assumption that (P1, P2,..., P25, P.) is a Hamiltonian cycle. Then P, and pas are not adjacent because both are odd. Since P2 is even by the same observation, P3 must be odd. Therefore the graph does not have a Harniltonian cycle. Repeating this process we obtain that pus is odd. Suppose that the graph has a Hamiltonian cycle. Since any vertex can be a starting point of a Hamiltonian cycle, we can find a Hamiltonian cycle (P1P2, P2, P.) with P1 = 1. Since P1 is odd, by the above observation, P2 must be even