The Colin de Verdière number and sphere representations of a graph


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    E. Andre'ev: On convex polyhedra in Lobachevsky spaces,Mat. Sbornik, Nov. Ser.,81 (1970), 445–478.

    Google Scholar 

  2. [2]

    R. Bacher andY. Colin de Verdière: Multiplicités des valeurs propres et transformations étoile-triangle des graphes,Bull. Soc. Math. France,123 (1995), 101–117.

    Google Scholar 

  3. [3]

    Y. Colin de Verdière: Sur un nouvel invariant des graphes et un critère de planarité,J. Combin. Theory B,50 (1990) 11–21.

    Google Scholar 

  4. [4]

    Y. Colin de Verdière: On a new graph invariant and a criterion for planarity, in:Graph Structure Theory (Robertson and P. D. Seymour, eds.), Contemporary Mathematics, Amer. Math. Soc. Providence, RI (1993), 137–147.

    Google Scholar 

  5. [5]

    H. van der Holst: A short proof of the planarity characterization of Colin de Verdière,J. Combin. Theory B,65 (1995) 269–272.

    Google Scholar 

  6. [6]

    H. van der Holst, L. Lovász andA. Schrijver: On the invariance of Colin de Verdière's graph parameter under clique sums,Linear Algebra and its Applications,226–228 (1995), 509–518.

    Google Scholar 

  7. [7]

    P. Koebe: Kontaktprobleme der konformen Abbildung,Berichte über die Verhandlungen d. Sächs. Akad. d. Wiss., Math.-Phys. Klasse,88 (1936) 141–164.

    Google Scholar 

  8. [8]

    A. V. Kostochka: Kombinatorial Analiz,8 (in Russian), Moscow (1989), 50–62.

    Google Scholar 

  9. [9]

    P. Mani: Automorphismen von polyedrischen Graphen,Math. Annalen,192 (1971), 279–303.

    Google Scholar 

  10. [10]

    J. Pach, P. K. Agarwal:Combinatorial Geometry, Willey, New York, 1995.

    Google Scholar 

  11. [11]

    N. Robertson, P. Seymour andR. Thomas: Sachs' linkless embedding conjecture,J. Combin. Theory B 64 (1995), 185–227.

    Google Scholar 

  12. [12]

    J. Reiterman, V. Rödl andE. Šinajová Embeddings of graphs in Euclidean spaces,Discr. Comput. Geom.,4, (1989), 349–364.

    Google Scholar 

  13. [13]

    J. Reiterman, V. Rödl andE. Šinajová: Geometrical embeddings of graphs,Discrete Math.,74 (1989), 291–319.

    Google Scholar 

  14. [14]

    H. Sachs: Coin graphs, polydedra, and conformal mapping,Discrete Math.,134 (1994), 133–138.

    Google Scholar 

  15. [15]

    O. Schramm: How to cage an egg,Invent. Math.,107 (1992), 543–560.

    Google Scholar 

  16. [16]

    M. Stiebitz: On Hadwiger's number—a problem of the Nordhaus-Gaddum type,Discrete Math,101 (1992), 307–317.

    Google Scholar 

  17. [17]

    W. Thurston:Three-dimensional Geometry and Topology, MSRI, Berkeley, 1991.

    Google Scholar 

  18. [18]

    W. Whiteley: Infinitesimally rigid polyhedra,Trans. Amer. Math. Soc.,285 (1984), 431–465.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kotlov, A., Lovász, L. & Vempala, S. The Colin de Verdière number and sphere representations of a graph. Combinatorica 17, 483–521 (1997).

Download citation

Mathematics Subject Classification (1991)

  • 05C