Akbari, Ghodrati, Hosseinzadeh (2017), *On the structure of graphs having a unique k-factor*, Aust. J. Combin. ([pdf](https://ajc.maths.uq.edu.au/pdf/69/ajc_v69_p063.pdf)) show:



> 
> ... we prove that there is no r-regular graph (r≥2) with a unique perfect matching.
> 
> 
> 


It seems natural to explore the stronger case:


**Question**: Does there exist an r-regular graph (r≥2) with a unique ***maximum*** matching?


I.e., the same problem with "perfect" replaced by "maximum", thereby including graphs without perfect matchings.


The answer is clearly "no" for r=2 (such graphs are unions of cycles; we can rotate any cycle to give a distinct maximum matching). For r=3, the answer is "no" for bridgeless cubic graphs, which have a perfect matching (by [Petersen's Theorem](https://en.wikipedia.org/wiki/Petersen%27s_theorem)) and therefore the above result applies. I'm not sure beyond this.



Does there exist an r-regular graph (r≥2) with a unique maximum matching?
Take a maximum matching $M$ and a vertex $v$ not in $M$. If $v$ has a neighbour $w$ not in $M$, then $M+vw$ is a larger matching. So $v$ has a neighbour $x$ which is in an edge $xy$ of $M$. Now $M-xy+vx$ is another maximum matching.


