Let G be a simple undirected graph.
If G is connected, and every vertex has degree $2$, then G is hamiltonian. (In fact, G only consists of the hamilton-circle)
Are there some weaker sufficient conditions for G being hamiltonian, if G is $2$-connected ? I am particular interested in the case, where all the degrees of the vertices are $2$ or $3$.
It would also be great, if someone knows necessary conditions for a $2$-connected graph to be hamiltonian.
With hamiltonian it is meant that G has a hamilton-circle, not just a hamilton-path.