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Question

\begin{aligned} \text { Show that there is no Hamiltonian path in } \\ & \operatorname { Cay } \left( \{ ( 1,0 ) , ( 0,1 ) \} : Z _ { 3 } \oplus Z _ { 2 } \right) \\ \text { from } ( 0,0 ) \text { to } ( 2,0 ) \end{aligned}

Solution

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Step 1
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Since $(2,0)+(1,0)=(0,0)$ if there was a Hamiltonian path from $(0,0)$ to $(2,0)$ then $Cay(\{(1,0),(0,1)\}:\mathbb Z_3\oplus\mathbb Z_2)$ would have a Hamiltonian circuit, which would contradict $\textbf{theorem 30.1}$ and the fact 2 and 3 are relatively prime.

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