Consider an undirected graph $G=(V,E)$, where $V=\{1,2,\ldots, n\}$, and initially $E=\varnothing$. Now take the following steps:
In the $1$st round, add undirected edges $(1,2)$, $(2,3)$, $\ldots$, $(n,1)$ to $E$;
In the $2$nd round, add undirected edges $(1,3)$, $(2,4)$, $\ldots$, $(n,2)$ to $E$;
In the $k$-th round, add undirected edges $(1,k+1)$, $(2,k+2)$, $\ldots$, $(n,k)$ to $E$;
If $k<[\frac{n}{2}]$, can we show that after the $k$-th round, the resulting graph $G$ contains $k$ edge-disjoint Hamiltonian cycles? If not, can you give a counter-example, and show me the correct way to derive $k$ edge-disjoint Hamiltonian cycles.
Thanks!