It is known that every $r$-regular graph on $2r+1$ vertices is Hamiltonian (Nash-Williams theorem, see here).
Now, I wonder if there is a simpler way to show that the graph on $4n+3$ ($n \ge 1$) vertices with degree sequence $(2n+1, \ldots, 2n+1, 2n+2)$ is Hamiltonian?
("a simpler way" means not to use the arguments in the proof of Nash-Williams theorem here).
For example, the graph with degree sequence $(3, 3, 3, 3, 3, 3, 4)$ must be Hamiltonian.