Let $G(V,E)$ be a graph with a [1-factorizations](https://en.wikipedia.org/wiki/Graph_factorization) $M$ and $m=|M|$ 1-factors. I am searching for graphs with **unique** 1-factorizations (i.e. there is only one 1-factorization).


Examples:


* [Cyclic graph](https://en.wikipedia.org/wiki/Cycle_graph) $C\_n$ with even $n$, with $m=2$ 1-factors.
* [Complete graph](https://en.wikipedia.org/wiki/Complete_graph) $K\_4$, with $m=3$ 1-factors.


(Some graphs have unique 1-factorizations under isomorphism, such as the [3-prism graph](https://en.wikipedia.org/wiki/Prism_graph) $Y\_3$, but I am not interested in those.)


1. Are there other examples?
2. What is the maximum number $m$ of 1-factors, when $G$ has a unique 1-factorization?


I conjecture that the answer to question 2 is $m=3$ (for $K\_4$), but I don't know how to prove that.


I would be very happy for every hint to an answer or a relevant literature.



Graphs with unique 1-Factorization
If there is only one edge colouring with $k$ colours of a graph with chromatic index $k$, the graph is said to be "uniquely edge colourable". If you search on that phrase (with and without the second "u") you will find that the problem is trivial for $k\le 2$, solved for $k\ge 4$ (only case $K\_{1,k}$) and for $k=3$ can be reduced to the case of cubic graphs. For example, see [this paper](http://www.sciencedirect.com/science/article/pii/S0167506008705119) of Andrew Thomason. It is said that an example is the generalized Petersen graph $P(9,2)$.


