$$

Decomposition

Into maximum matchings

• If $m$ is odd you can decompose it into $m$ matchings of size $\frac{m-1}{2}$: visualize it so that the vertices form a regular $m$-gon, and group the edges by slope.
• If $m$ is even, use the solution above for the $m-1$ first vertices, and add the only possible missing edge to each of them.

Into Hamiltonian cycles

Still about grouping by slopes: