3-Perfect hamiltonian decomposition of the complete graph (Q2848739)

Let \(K_n\) be the complete graph on \(n\) vertices and let \(i\) be an integer with \(2\leq i\leq (n-1)/2\). A Hamiltonian decomposition \(\mathcal H\) of \(K_n\) is called \textit{\(i\)-perfect} if the set of the chords at distance \(i\) of the Hamiltonian cycles in \(\mathcal H\) is the edge set of \(K_n\). It is demonstrated that there exists a \(3\)-perfect Hamiltonian decomposition of \(K_n\) for all odd \(n\geq 7\).