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Proof of Toft’s Conjecture: Every Graph Containing No Fully Odd K 4 Is 3-Colorable

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Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

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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.

This work was supported in part by the Air Force Office of Scientific Research under grant F49620-93-1-0041 awarded to Rutgers Cent

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Hong Kong, Hong Kong

    Wenan Zang

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  1. Wenan Zang
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Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

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© 1998 Springer-Verlag Berlin Heidelberg

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Zang, W. (1998). Proof of Toft’s Conjecture: Every Graph Containing No Fully Odd K 4 Is 3-Colorable. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_30

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  • DOI: https://doi.org/10.1007/3-540-68535-9_30

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  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

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