Smaller Cycle Inside Hamiltonian Graph

Question

It seems obvious to me that a cycle $C_k$ in a graph on $n$ vertices ($n > k$) cannot feature in a Hamiltonian cycle for the graph, but I'm having trouble writing it down in a rigorous fashion. Any ideas as how to make this watertight? I'm an applied mathematician, save me :) .

Answer 1

I'm assuming "feature" means "occurs in". As you said, it's obvious, but the formality of writing a proof can be constraining. I'll try a proof by contradiction (I understand some find that such proofs obfuscate what you're trying to prove, but here you're proving something that is intuitively obvious, so, to me at least, it feels more natural to show how it would be observe if you could have such a cycle).

Say we have a Hamiltonian cycle $H$ in a graph $G$ where $|V(G)|=n$ and a cycle $C$ of length $k<n$ where $C$ is contained in $H$. If the initial/terminal vertex of $C$ is not the same as that of $H$, then $H$ cannot be a cycle, as only its initial and terminal vertex can (and must) be equal. (Obviously, the initial vertex of a cycle is arbitrary, but we can fix one). Otherwise, say the initial/terminal of $C$ is equal to that of $H$. Since $H$ contains $C$ and repeats only its initial/terminal vertex, then $H=C$. But then $H$ is not Hamiltonian, as it has length $k<n$.