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On Reed’s Conjecture about ω,Δ and χ

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Graph Theory in Paris

Part of the book series: Trends in Mathematics ((TM))

Abstract

For a given graph G, the clique number ω(G), the chromatic number χ(G) and the maximum degree Δ(G) satisfy ω(G) ≤ χ(G) ≤ Δ(G)+1. Brooks showed that complete graphs and odd cycles are the only graphs attaining the upper bound Δ(G)+1. Reed conjectured \( \chi (G) \leqslant \left\lceil {\tfrac{{\Delta + 1 + \omega }} {2}} \right\rceil \) . In this paper we will present some partial solutions for this conjecture.

Parts of this research were performed within the RIP program (Research in Pairs) at the Mathematisches Forschungsinstitut Oberwolfach. Hospitality and financial support are gratefully acknowledged.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Universität zu Köln, D-50969, Köln, Germany

    Bert Randerath

  2. Institut für Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, D-09596, Freiberg, Germany

    Ingo Schiermeyer

Authors
  1. Bert Randerath
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  2. Ingo Schiermeyer
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Editor information

Editors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne cedex, France

    Adrian Bondy

  2. Equipe Combinatoire et Optimisation Case 189, Université Pierre et Marie Curie, Paris 6, 75252, Paris Cedex 05, France

    Jean Fonlupt , Jean-Claude Fournier  & Jorge L. Ramírez Alfonsín ,  & 

  3. Laboratoire d’Informatique Fondamentale d’Orléans, Bâtiment IIIA, Rue Léonard de Vinci, B.P. 6759, 45067, Orléans Cedex 2, France

    Jean-Luc Fouquet

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Randerath, B., Schiermeyer, I. (2006). On Reed’s Conjecture about ω,Δ and χ . In: Bondy, A., Fonlupt, J., Fouquet, JL., Fournier, JC., Ramírez Alfonsín, J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7400-6_26

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