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

Extended Abstract
  • Wenan Zang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)


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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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Wenan Zang
    • 1
  1. 1.Department of MathematicsUniversity of Hong KongHong Kong

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