Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.Comment: 18 pages, many images; final versio

Toward a Language Theoretic Proof of the Four Color Theorem, arXiv:1006.1324v1, preprint

Abstract. This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem

Varieties of nilpotent elements for simple Lie algebras I:Good primes

Let G be a simple algebraic group over k = ℂ, or F̄p where p is good. Set g = Lie G. Given r ∈ ℕ and a faithful (restricted) representation ρ: g → gl(V), one can define a variety of nilpotent elements Nr,ρ(g) = {x ∈ g: ρ(x)r = 0}. In this paper we determine this variety when ρ is an irreducible representation of minimal dimension or the adjoint representation