Proof of Toft’s Conjecture: Every Graph Containing No Fully Odd K4 Is 3-Colorable
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.
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