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May 30, 2018 at 16:48 comment added Lisa Vibbert Thank you for your advice. I neglected to consider that the discrete partition can be added to any klatsch without invalidating the conditions, so my example is wrong in that there are actually two of them on a three element set up to isomorphism. (I ran into to this problem when trying to design asynchronous arbiter mesh networks, where the discrete partition would correspond to a layer that doesn't do anything.)
May 28, 2018 at 18:40 comment added Gerhard Paseman This setup also generalizes partitioning a complete graph into disjoint matchings, or disjoint unions of triangles. I suggest seeing if there are algorithms to generate these special K, and see if citing literature does any thing more. Gerhard "Is Not Talking Breakfast Cereal" Paseman, 2018.05.28.
May 28, 2018 at 18:32 history answered Gerhard Paseman CC BY-SA 4.0