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Aug 29, 2017 at 17:50 history edited Peter Heinig CC BY-SA 3.0
Corrected incorred cyclic shifting in the examples I gave. Adde a warning that this is a wasteful construction.
Aug 29, 2017 at 17:49 comment added Ben Barber The graph picture for anyone who doesn't want to work out the details: this is a Hamilton cycle of length $2k$ together with a star whose leaves are all the elements of one colour class and whose centre is in the other. Any choice of matching edge for the centre of the star is a chord that splits the cycle into two paths, each of which has a unique perfect matching, giving the $k$ options.
Aug 29, 2017 at 17:49 vote accept Timothy Chow
Aug 29, 2017 at 17:44 history edited Peter Heinig CC BY-SA 3.0
Corrected incorred cyclic shifting in the examples I gave.
Aug 29, 2017 at 17:44 comment added Timothy Chow Nice construction! Shortly after posting my question, I discovered the following paper by Kim, Lee, and Seol that also answers the question: ijpam.eu/contents/2005-19-3/12/12.pdf
Aug 29, 2017 at 17:36 history answered Peter Heinig CC BY-SA 3.0