Connectivity Preserving Hamiltonian Cycles in k-Connected Dirac Graphs

We show that for \(k \ge 2\) $k\ge 2$ , there exists a function \(f(k) = O(k)\) $f(k)=O(k)$ such that every k-connected graph G of order \(n \ge f(k)\) $n\ge f(k)$ with minimum degree at least \(\frac{n}{2}\) $\frac{n}{2}$ contains a Hamiltonian cycle H such that \(G-E(H)\) $G-E(H)$ is k-connected. We also show that for \(k \ge 2\) $k\ge 2$ and \(\ell \ge 2\) $\ell \ge 2$ , there exists a function \(g(k,\ell ) = O(k\ell )\) $g(k,\ell )=O(k\ell )$ such that every k-connected graph G of order \(n \ge g(k,\ell )\) $n\ge g(k,\ell )$ with minimum degree at least \(\frac{n}{2}\) $\frac{n}{2}$ contains \(\ell \) $\ell$ edge-disjoint Hamiltonian cycles \(H_1,H_2,\ldots ,H_\ell \) $H_1,H_2,\dots ,H_{\ell }$ such that \(G-\cup _{1 \le i \le \ell }E(H_i)\) $G-{\cup }_{1\le i\le \ell }E(H_i)$ is k-connected. Furthermore, a similar result with an improved lower bound on n is shown when the connectivity of G is exactly k.