Independence, odd girth, and average degree

  • Authors:

  • Christian Löwenstein,;Anders Sune Pedersen,;Dieter Rautenbach,;Friedrich Regen

  • Affiliations:

  • Institut für Mathematik, TU Ilmenau Postfach 100565, D-98684 Ilmenau, Germany;Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55 DK-5230 Odense M, Denmark;Institut für Mathematik, TU Ilmenau Postfach 100565, D-98684 Ilmenau, Germany;Institut für Mathematik, TU Ilmenau Postfach 100565, D-98684 Ilmenau, Germany

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2011

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Abstract

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n−m−1)/7. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96-111, 2011 © 2011 Wiley Periodicals, Inc.