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

Examples:

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

I have three questions:

- How are such graphs called?
- Are there other examples than $C_n$ and $K_4$?
- 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.

**Edit2:** The answer to this question I have used in a recent article in Physical Review Letters, where I cite this MO question as reference [24]. See Figure 2 for a detailed variant of the application of Ilya's answer. Thanks Ilya!

manydistinct perfect matchings, in particular question 3. But i suspect m=3 is the limit. I should specify that, thank you! $\endgroup$ – Mario Krenn Apr 12 '17 at 13:59disjointif they have empty intersection; they aredistinctif they are different. $\endgroup$ – David Richerby Apr 12 '17 at 22:21