Let $G(V,E)$ be a graph. I am searching for graphs with only **distinct** [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 $m=2$ distinct perfect matchings.
 - [Complete graph][3] $K_4$, with $m=3$ distinct 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 of perfect matchings, if the graph has only completly distict perfect matchings?

For question 3, it seems to me that $K_4$ with 3 different, distinct 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!

  [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