Proof of toft’s conjecture: Every graph Containing No Fully Odd K 4 is 3-colorable

Abstract

The graph 3-coloring problem arises in connection with certain scheduling and partition problems. As is well known, this problem is NP-complete and therefore intractable in general unless NP = P. The present paper is devoted to the 3-coloring problem on a large class of graphs, namely, graphs containing no fully odd K 4, where a fully odd K 4 is a subdivision of K 4 such that each of the six edges of the K 4 is subdivided into a path of odd length. In 1974, Toft conjectured that every graph containing no fully odd K 4 can be vertex-colored with three colors. The purpose of this paper is to prove Toft’s conjecture.