# Is there a graph with 2-factor that is not hamiltonian?

If a graph $$G$$ has a 2-factor it means it is a 2-regular subgraph that contains all vertices of $$G$$. Isn't it a hamiltonian cycle? because it is 2-regular so it is a cycle and it contains all vertices of the graph.

A $$2$$-factor might not be connected; it might consist of multiple cycles that, together, include all the vertices. The standard example is the Petersen graph:
• However, it does have a $$2$$-factor: take the "outside" $$5$$-cycle together with the "inside" $$5$$-cycle, for example.