This is the ninth in a series, the first being here: Part 1, and the previous here: Part 8.

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.

Part 10

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