Non-Hamiltonian $k$-connected $k$-regular graphs ($k>3$)? The Petersen graph provides us with an example of an non-Hamiltonian 3-connected 3-regular graph. Are any 4-connected 4-regular graphs known to be non-Hamiltonian? What about generic $k$-connected $k$-regular graphs?

According to this paper (Regular n-Valent n-Connected NonHamiltonian Non-n-Edge-Colorable Graphs by G. H. J. Meredith) the answer is yes (by construction) for $k=4$ and $k \ge 8$ but the other cases are not shown there. You can see a nice picture of the construction (which goes via the Petersen graph) for $k=4$ on page 5.
