Here is a Q1 graph for n=13
o / o-o-o-o-o-o-o-o-o-o \ o-o
The corresponding Q1 partition is 5+5+3. Unfortunately this does not generalize. For instance, the following graph:
o / o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o \ o-o
prohibits the partitions 5+5+3+3+3 and 6+5+5+3, so it is Q2.
So far we have found Q1 graphs (or proven that they don\’t exist) of all orders n up through 18. I have made very little progress with n=19. If anyone finds an order 19 Q1 graph, please let me know about it. I do have some more things to say about Q1 graphs and partitions, including work and suggestions by my friend Sean (as well as some more Q1 graphs). In my next post I will start to discuss those issues.