Definitions A cubic graph (simple) $G$ is a 3-regular graph. An edge cut $K$ is cycle separating if $G-K$ is disconnected and at least two components of $G-K$ have circuits. A graph is cyclically edge-$k$-connected if no set of fewer than $k$ edges is cycle separating. The cyclical edge connectivity of $G$ is the largest integer $k$ such that the graph is cyclically edge-$k$-connected.
Our question is the following:
Question Would anybody know any examples of cubic graphs which have cyclically edge connectivity at least 5 which do not have Hamiltonian paths (not cycles)? If there is such an example and it is edge-3-colorable, would you know of an example which is not edge-3-colorable?