Short proofs of coloring theorems on planar graphs

A lower bound on the number of edges in a k -critical n -vertex graph recently obtained by Kostochka and Yancey yields a half-page proof of the celebrated Grötzsch Theorem that every planar triangle-free graph is 3-colorable. In this paper we use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among which is the Grünbaum–Aksenov Theorem that every planar graph 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 4 is 3 -colorable.