I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph is 5-colorable. It's rather simple to prove Five Color Theorem in a standard mathematical approach (without resort to computers) and many graph theory textbooks explain this proof based on induction. (Even the wikipedia article covers the proof.)

Now, I'm interested in an **algebraic proof** of Five Color Theorem using chromatic polynomials. Let $\pi_G(x)$ be a chromataic polynomial of a graph $G$. The Five Color Theorem is proved once $\pi_G(5) > 0$ is shown for any planar graph $G$ because $\pi_G(5)$ is the number of 5-colorings of a graph $G$. In fact, Birkhoff introduced the chromatic polynomial with the hope of proving the Four Color Theorem by showing that $\pi_G(4) > 0$ for any planar graph. This approach was in vain, yet he and Lewis succeeded in showing that $\pi_G(x) > 0$ for any planar graph $G$ when $x$ is a real number larger than or equal to 5. For this algebraic proof of Five Color Theorem, many people point to 97-page paper written by Birkhoff and Lewis in 1946.

As a computer science TA of an advanced graph theory course, I think this algebraic proof is worth for a brief explanation and may be an adequate thesis topic for master student or even an undergraduate student. But it's somewhat hard to find out which part of the 97 page document is about an algebraic proof of Five Color Theorem. (Probably it's because I'm not a native speaker and not familiar with paper writing styles almost 70 years ago.)

- Is there anyone who already read Birkhoff and Lewis and can tell me which section is about the actual algebraic proof?
- Is there any other
*recent*lecture notes or textbooks explaining this algebraic proof in a concise fashion?