First-Order Definability of Trees and Sparse Random Graphs

  • Authors:

  • TOM BOHMAN;ALAN FRIEZE;TOMASZ ŁUCZAK;OLEG PIKHURKO;CLIFFORD SMYTH;JOEL SPENCER;OLEG VERBITSKY

  • Affiliations:

  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu);Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu);Department of Discrete Mathematics, Adam Mickiewicz University, Poznań 61-614, Poland (e-mail: tomasz@amu.edu.pl);Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@moser.math.cmu.edu, alan@random.math.cmu.edu, pikhurko@cmu.edu);Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: csmyth@math.mit.edu);Courant Institute, New York University, New York, NY 10012, USA (e-mail: spencer@cims.nyu.edu);Institut für Informatik, Humboldt Universität Berlin, D-10099 Berlin, Germany (e-mail: verbitsk@informatik.hu-berlin.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2007

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Abstract

Let D(G) be the smallest quantifier depth of afirst-order formula which is true for a graph G but falsefor any other non-isomorphic graph. This can be viewed as a measurefor the descriptive complexity of G in first-orderlogic.We show that almost surely D(G)=θ(lnn / ln lnn), where G is a random tree of order n or thegiant component of a random graph G(n,c/n)with constant cD(T) for a tree T of order n andmaximum degree l, so we study this problem as well.