The **arrangement graph** \(A_{n,k}\) (with \(n>k\geq 1\)) has a vertex set consisting of all possible permutations of \(k\) elements chosen from the ground set of \(n\) elements \(\{1,2,\ldots , n\}\). Two vertices (nodes) \(u\) and \(v\) of \(A_{n,k}\) are adjacent if their corresponding permutations differ in exactly one of the \(k\) positions.

A **hamiltonian cycle** in a graph is a closed path through a graph that visits each node exactly once.

Find three edge disjoint hamiltonian cycles in \(A_{5,2}\) shown below.

Figure 1: The graph \(A_{5,2}\) contains three edge disjoint hamiltonian cycles.

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