Progress on the Four-Color Theorem

Abstract

The four-color theorem, that every loopless planar graph is 4-colorable, was proved in 1977 by Appel and Haken, but there remain open several conjectured extensions. Here we report on a new (but still computer-based) proof of the four-color theorem itself, and on some progress towards two of the open extensions. We also survey some related results on minors of graphs and on linkless embeddings of graphs in 3-space.