Georgia Institute of Technology College of Sciences

A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Grotzsch Theorem that every planar triangle-free graph is 3-colorable. We use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Grunbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable. Joint work with O. Borodin, A. Kostochka and M. Yancey.