The question is:
Does there exist a simple connected undirected graph $G$ with $7$ vertices with minimal degree $3$ but does not contain any hamiltonian cycle?
I've been trying to find an example for quite long time, but I still cannot think of one. The restriction "minimal degree 3" is giving me an headache, since I can always find a graph with no-hamiltonian cycle with "almost minimal degree 3", but whenever one edge is added so to satisfy the condition, it becomes hamiltonian...
So the question comes. Is there even a single graph with above properties? Maybe I am being a bit un imaginative, but I've found questions about finding non-hamiltonian graphs with certain properties quite hard so far.
It would be great if you could explain your strategy too with an example, since I seem to lack what is needed in this kind of exercise: I can't get the "feel" of it.
Thanks in advance,