Let $G(V,E)$ be a graph. I am searching for graphs with only **disjoint** [perfect matchings][1] (i.e. every edge only appears in at most one of the perfect matchings).

Examples:

 - [Cyclic graph][2] $C_n$ with even $n$, with $m=2$ disjoint perfect matchings.
 - [Complete graph][3] $K_4$, with $m=3$ disjoint perfect matchings.

I have three questions:

 1. How are such graphs called?
 2. Are there other examples than $C_n$ and $K_4$?
 3. What is the maximum number $m$ of perfect matchings, if the graph has only completly disjoint perfect matchings?

For question 3, it seems to me that $K_4$ with $m=3$ different, disjoint perfect matchings is the optimum, but I have no proof for that.

Every hint to an answer or to relevant literature would be very much appreciated!

**Edit:** I am interested in undirected graphs only for the moment.

  [1]: https://en.wikipedia.org/wiki/Matching_(graph_theory)
  [2]: https://en.wikipedia.org/wiki/Cycle_graph
  [3]: https://en.wikipedia.org/wiki/Complete_graph