In my seventh grades, we’ve been investigating the traceability of graphs. Today in one of my classes we were talking through some of their conjectures and spent some time thinking about the relationship between traceability and 2-colorability. (Some of the Vi Hart fans had the latter notion in hand from this video.) The conjecture on the table was that if a graph is not 2-colorable, then it is not traceable.
We thought we found a counterexample to the converse—or, the “other way around”—but then modified/corrected our notion or 2-colorability so that the exterior of the shape also had to be a color. This negated our counterexample.
As often happens in mathematics, today ended in a “to be continued.” Things were a little loosey-goosey—this has definitely not been one of my more directed or planned investigations. But good things came out and they were pretty into it. We’ll see how we pull it all together.
Also, I had an awesome time with one of my independent study students today. We followed up on our previous conversation about cardinality, and talked some about long division and different bases. We didn’t make it to Cantor’s diagonalization argument, because he wanted to talk about, well, a lot of other things. But in particular, this video that he found of William Thurston that I don’t think I’d run across before. Yay!
No need to rush—we’ll get to play more next week. 🙂