Is there a (finite, connected) 3-regular graph with no Hamiltonian path?
1) The Petersen graph has no Hamiltonian cycle, but it does have a Hamiltonian path.
2) In lieu of https://en.wikipedia.org/wiki/Lov%C3%A1sz_conjecture, the counter example (if it exists) is probably not vertex transitive.
3) If $|G| = n$, and if there is a independent set of size $ > n/2$ then there is no Hamiltonian path. However, no example can be constructed using this observation, because of: Maximum independence number of any $d$-regular graph on $v$ vertices
I'm also curious about the $k$-regular case. (For $k \geq 3$.)