A graph G is tough if the number of $c(G-s) \le |S|$ for all $S \in V(G)$.
And I read that every tough graph is Hamiltonian, but the other way is not true. I was wondering if there is an example of a graph that is tough but is not Hamiltonian.
I think Petersen's graph is one example, but are there any other such graphs?
Any help would be appreciated.
Chvátal's original paper defining toughness gives an example of a $7$-vertex graph which is tough but not Hamiltonian:
This is easy to verify, since a Hamiltonian cycle would have to use both edges out of each of the vertices with degree $2$, which includes three edges out of the center vertex.