I had my fifth evening of math with my kids. I got a box of 100 washers from the hardware store and some yarn. I had the kids draw some graphs and then make some graphs with the washers as nodes and the yarn as the edges. I let them make some of thier own, then I had them make these:
The square:
o-o | | o-o
and the claw:
o | o / \ o o
and asked them if they were the same (“What do you mean, Dad? Of course they’re not the same.”)
Then I had them make the pentagon and the pentagram. I asked if they were the same (again “of course not”). So I took two copies of the square. “Are they the same?” “Of course.” “Well then, why aren’t the square and the claw the same?” With time they answered that one had a node that was hooked to three other nodes rather than just two. Next they spent some time trying to get the pentagon to look like the pentagram. I suggested they try to work with the pentagram itself. B picked it up and it immediately fell apart into the pentagon.
Next I showed how I could cut two edges of the pentagon to get two connected graphs, one with two nodes and the other with three, thus making the partition of 5: 3+2. I then asked how many partitions of 4 they could make with the claw. They discovered that the only one that they couldn’t make was 2+2. I asked this question.
Q: How many partitions of 4 can’t you make with the claw?
The answer is 1, so we call the claw a Q1 graph.
A couple nights later I had them look for more Q1 graphs. B tried one that he called the chicken claw:
o | o | o / \ o o / \ o o
He quickly figured out that this is not a Q1 graph. In the mean time A worked on what she called the ice cream cone:
o-o-o | / o o |/ o
But she wasn’t able to finish her analysis of it inthe time we had.