A Graph in which subgraph is on Hamiltonian is Hamiltonian?

Question

If a Graph has a sub graph which is not Hamiltonian, Will the Original graph also non Hamiltonian?

For Example, K3,4 is not Hamiltonian. What is I connect 10 K3,4 graphs in a way to makeup Meredith Graph? Can I say since K3,4 is on Hamiltonian, the super graph is also not?

Answer 1

No way!

Note the fallacy in your reasoning. Pick $C_5 $ which is the $5$-cycle. Note that it is Hamiltonian.

Now consider the following subgraph of $C_5$

Is it Hamiltonian?

Note that we can form a non-Hamiltonian sub-graph given any Hamiltonian graph. Hence the "Hamiltoniancy" of a subgraph does not affect the super-graph at all.