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Combinatorica

, Volume 17, Issue 4, pp 483–521 | Cite as

The Colin de Verdière number and sphere representations of a graph

  • Andrew Kotlov
  • László Lovász
  • Santosh Vempala
Article

Abstract

Colin de Vedière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G)≤2 if and only ifG is outerplanar, and μ(G)≤3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then μ(G)≥n−4, and if it is planar, then μ(G)≥n−5. We give a full characterization of maximal planar graphs whose complementsG have μ(G)=n−5. In the opposite direction we show that ifG does not have “twin” nodes, then μ(G)≥n−3 implies that the complement ofG is outerplanar, and μ(G)≥n−4 implies that the complement ofG is planar.

Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.

Mathematics Subject Classification (1991)

05C 

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Copyright information

© János Bolyai Mathematical Society 1997

Authors and Affiliations

  • Andrew Kotlov
    • 1
  • László Lovász
    • 3
  • Santosh Vempala
    • 2
  1. 1.Dept. of Mathematics, YaleUniversityNew Haven
  2. 2.Dept. of Computer ScienceCarnegie Mellon UniversityPittsburgh
  3. 3.Dept. of Computer Science, YaleUniversityNew Haven

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