The Colin de Verdière number and sphere representations of a graph

Andrew Kotlov, L. Lovász, Santosh Vempala

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

Colin de Verdière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G) ≤ 2 if and only if G is outerplanar, and μ(G) ≤ 3 if and only if G is planar. We prove that if the complement of a graph G on n 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 complements G have μ(G) = n - 5. In the opposite direction we show that if G does not have "twin" nodes, then μ(G) ≥ n - 3 implies that the complement of G is outerplanar, and μ(G) ≥ n - 4 implies that the complement of G 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.

Original languageEnglish
Pages (from-to)483-521
Number of pages39
JournalCombinatorica
Volume17
Issue number4
Publication statusPublished - 1997

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Complement
Planar graph
Graph in graph theory
If and only if
Imply
Outerplanar Graph
Invariant
Vertex of a graph
Formulation

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

Cite this

The Colin de Verdière number and sphere representations of a graph. / Kotlov, Andrew; Lovász, L.; Vempala, Santosh.

In: Combinatorica, Vol. 17, No. 4, 1997, p. 483-521.

Research output: Contribution to journalArticle

Kotlov, A, Lovász, L & Vempala, S 1997, 'The Colin de Verdière number and sphere representations of a graph', Combinatorica, vol. 17, no. 4, pp. 483-521.
Kotlov, Andrew ; Lovász, L. ; Vempala, Santosh. / The Colin de Verdière number and sphere representations of a graph. In: Combinatorica. 1997 ; Vol. 17, No. 4. pp. 483-521.
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