Partitions of Graphs IX: The case of 13

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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One Response to Partitions of Graphs IX: The case of 13

  1. Pingback: Partitions of Graphs X: The Case of Composite Integers « MATH with my KIDS

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