Short proofs of coloring theorems on planar graphs

A recent lower bound on the number of edges in a k-critical nvertex graph 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 whose is the Grünbaum-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. ∗Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia E-mail: [email protected]. Research of this author is supported in part by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research. †University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA and Sobolev Institute of Mathematics, Novosibirsk 630090, Russia. E-mail: [email protected]. Research of this author is supported in part by NSF grant DMS-0965587. ‡University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA, E-mail: [email protected] §University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA, E-mail: [email protected]. Research of this author is partially supported by the Arnold O. Beckman Research Award of the University of Illinois at Urbana-Champaign and from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.”