# Article

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Summary:
For any \$n\ge 1\$ and any \$k\ge 1\$, a graph \$S(n,k)\$ is introduced. Vertices of \$S(n,k)\$ are \$n\$-tuples over \$\lbrace 1, 2, \ldots , k\rbrace \$ and two \$n\$-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs \$S(n,3)\$ are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of \$S(n,k)\$. Together with a formula for the distance, this result is used to compute the distance between two vertices in \$O(n)\$ time. It is also shown that for \$k\ge 3\$, the graphs \$S(n,k)\$ are Hamiltonian.
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