General Math Stuff

[Surlethe]

Sorry, habit. The $---$ sets off math in LaTeX. Try copying what I write inside dollar signs into something like this, and you'll see what I mean. For example, try $\int_{\partial M}\omega = \int_M d\omega$.

[Grog]

Could do better: there exists a non-planar 2-colorable graph. Spoiler

Take the complete 5-vertex graph, color every vertex red, and for each edge, put an extra vertex, thus subdividing each into two edges. Color each new vertex blue. The graph is non-planar by Kuratowski's theorem.

Well the complete bipartite graph on 6 vertices, 3 in each part, is not planar by the same theorem.

[Grog]

I have thought a fair bit about graph colorings, I have a paper about it actually. Right now I cant come up with a nice answer for the question about russia tough, I might just be stupid today.

[Grog]

Some nice exercises. Prove that all planar graphs are 5 colorable (this is a bit tricky but fun) easier but still fun prove that they are 6 colorable.

A nice fact about planar graphs is that they are 3 colorable if they contain at most 3 triangles.