Building a non-hamiltonian graph of $p$ vertices of $\frac{p-1}2$ degree each.

Question

I want to build some graph with $p$ vertices all with degree of atleast $\frac{p-1}{2}$ that isn't hamiltonian.

I imagine this is possible, but I can't seem to do it. Any suggestions? Perhaps looking at the havel-hakimi theorem I should just keep creating $p$ length degree sequences of $\large\text{ceilling}(\frac{p-1}{2})$ and breaking it down into a graphical form and building it up using reverse $H-H$ is there some easier way? Thank you!

Answer 1

Hints:

(1) If $p$ is odd, consider a certain complete bipartite graph.

(2) If $p$ is even, then what you seek cannot exist...