7 research outputs found
Short proofs of coloring theorems on planar graphs
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 Gr\"otzsch
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 whose is the Gr\"unbaum-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.Comment: 13 pages, 4 figure