Hamiltonian path in countable connected graph such that $\text{deg}(v)=\omega$ for all $v$ [closed]

Question

Is there a countable connected graph $G=(\omega, E)$ such that $\text{deg}(v)=\omega$ for all $v\in\omega$, but there is no Hamiltonian path in $G$?

Answer 1

Take the natural numbers as vertices and connect a pair of numbers with an edge if their difference is divisible by three. Also connect 1 with 2 and 2 with 3.