The Chicken’s Claw

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.

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