Connectivity Preserving Hamiltonian Cycles in $k$-Connected Dirac Graphs

Abstract: We show that for $k \geq 2$, there exists a function $f(k) = O(k)$ such that every $k$-connected graph $G$ of order $n \geq f(k)$ with minimum degree at least $\frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. Applying Nash-Williams' result on edge-disjoint Hamiltonian cycles, we also show that for $k \geq 2$ and $\ell \geq 2$, there exists a function $g(k,\ell) = O(k\ell)$ such that every $k$-connected graph $G$ of order $n \geq g(k,\ell)$ with minimum degree at least $\frac{n}{2}$ contains $\ell$ edge-disjoint Hamiltonian cycles $H_1,H_2,\ldots,H_\ell$ such that $G-\cup_{1 \leq i \leq \ell}E(H_i)$ is $k$-connected. As a corollary, we have a statement that refines the result of Nash-Williams for $k$-connected graphs with $k \leq 8$. Moreover, when the connectivity of $G$ is exactly $k$, a similar result with an improved lower bound on $n$ can be shown, which does not depend on the result of Nash-Williams.