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Irreducible configurations and the four color conjecture

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Part of the Lecture Notes in Mathematics book series (LNM,volume 642)

Abstract

The classical four color reduction process takes on a new appearance in the light of the recently begun theory of open sets of colorings. In this paper we show that specific configurations and clusters can be simply classified as either reducible or irreducible, without appealing to the truth or falsity of the Four Color Conjecture (4CC). By treating irreducibility and reducibility together, we hope to round out the theory and gain a better understanding of why clusters do or do not reduce.

The new methods are illustrated by giving a condensed proof of irreducibility for all reasonable candidates on the order of the six-ring or less. The principle tools are the union and splicing properties of open sets, and the rotation of "antiset" pairs (Lemma 1). Complete details and extension to higher rings will come in a later paper or papers.

At the conclusion it is shown that the 4CC is equivalent to the set of irreducible clusters (in our definition) being infinite!

Keywords

  • Plane Graph
  • Interior Vertex
  • Minimal Graph
  • Proper Ring
  • Splice Diagram

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1978 Springer-Verlag Berlin Heidelberg

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Bernhart, F.R. (1978). Irreducible configurations and the four color conjecture. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070363

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  • DOI: https://doi.org/10.1007/BFb0070363

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08666-6

  • Online ISBN: 978-3-540-35912-8

  • eBook Packages: Springer Book Archive