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Even Maps, the Colin de Verdière Number and Representations of Graphs


Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on σ(G) which is, in general, tight.

Equality between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.

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  1. R. Bacher and Y. Colin de Verdière: Multiplicités des valeurs propres et transformations étoile-triangle des graphes, Bulletin de la Société Mathématique de France 123 (1995), 517–533.

    MathSciNet  Article  Google Scholar 

  2. Y. Colin de Verdière: Sur un nouvel invariant des graphes et un critère de planarité, Journal of Combinatorial Theory, Series B 50 (1990), 11–21.

    MathSciNet  Article  Google Scholar 

  3. Y. Colin de Verdière: On a new graph invariant and a criterion for planarity, in: N. Robertson and P.D. Seymour, editors, Graph Structure Theory, volume 147 of Contemporary Mathematics, pages 137–147. American Mathematical Society, 1991.

  4. V. Dujmovlć and S. Whitesides: Three-dimensional drawings, in: R. Tamassia, editor, Handbook of Graph Drawing and Visualization, Discrete Mathematics and Its Applications. CRC Press, 2013.

  5. G. Ewald and G. C. Shephard: Stellar subdivisions of boundary complexes of convex polytopes, Math. Ann. 210 (1974), 7–16.

    MathSciNet  Article  Google Scholar 

  6. J. Foisy: A newly recognized intrinsically knotted graph, J. Graph Theory 43 (2003), 199–209.

    MathSciNet  Article  Google Scholar 

  7. F. Goldberg: Optimizing Colin de Verdière matrices of K4,4Linear Algebra and its Applications 438 (2013), 4090–4101.

    MathSciNet  Article  Google Scholar 

  8. H. van der Holst: A Short Proof of the Planarity Characterization of Colin de Verdière, Journal of Combinatorial Theory, Series B 65 (1995), 269–272.

    MathSciNet  Article  Google Scholar 

  9. H. van der Holst, M. Laurent and A. Schrijver: On a Minor-Monotone Graph Invariant, Journal of Combinatorial Theory, Series B 65 (1995), 291–304.

    MathSciNet  Article  Google Scholar 

  10. H. van der Holst, L. Lovász and A. Schrijver: On the invariance of Colin de Verdière’s graph parameter under clique sums, Linear Algebra and its Applications 226 (1995), 509–517.

    MathSciNet  Article  Google Scholar 

  11. H. van der Holst, L. LOvÁsz and A. Schrijver: The Colin de Verdière graph parameter, 29–85, Bolyai Society Mathematical Studies. János Bolyai Mathematical Society, Hungary, 1999.

    MATH  Google Scholar 

  12. H. van der Holst and R. Pendavingh: On a graph property generalizing planarity and flatness, Combinatorica 29 (2009), 337–361.

    MathSciNet  Article  Google Scholar 

  13. I. Izmestiev: The Colin de Verdière number and graphs of polytopes, Israel Journal of Mathematics 178 (2010), 427–444.

    MathSciNet  Article  Google Scholar 

  14. V. Kaluža and M. Tancer: Even maps, the Colin de Verdière number and representations of graphs, in: Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, 2642–2657, 2020.

  15. V. Kaluža and M. Tancer: Even maps, the Colin de Verdière number and representations of graphs, arXiv:1907.05055, 2019.

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

    MathSciNet  Article  Google Scholar 

  17. L. Lovász: Steinitz representations of polyhedra and the Colin de Verdière number, Journal of Combinatorial Theory, Series B 82 (2001), 223–236.

    MathSciNet  Article  Google Scholar 

  18. L. Lovász and A. Schrijver: A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs, Proceedings of the American Mathematical Society 126 (1998), 1275–1285.

    MathSciNet  Article  Google Scholar 

  19. L. Lovász and A. Schrijver: On the null space of a Colin de Verdière matrix, Annales de l’Institut Fourier 49 (1999), 1017–1026.

    MathSciNet  Article  Google Scholar 

  20. R. McCarty: The extremal function and Colin de Verdière graph parameter, Electronic Journal of Combinatorics, 25(2):P2.32, 2018.

    Article  Google Scholar 

  21. R. M. McCarty: Personal communication, 2019.

  22. R. Pendavingh: On the Relation Between Two Minor-Monotone Graph Parameters, Combinatorica 18 (1998), 281–292.

    MathSciNet  Article  Google Scholar 

  23. N. Robertson, P. Seymour and R. Thomas: Sachs’ linkless embedding conjecture, Journal of Combinatorial Theory, Series B 64 (1995), 185–227.

    MathSciNet  Article  Google Scholar 

  24. N. Robertson and P. D. Seymour: Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), 65–110.

    MathSciNet  Article  Google Scholar 

  25. N. Robertson and P. D. Seymour: Graph minors. XX. Wagner’s conjecture, J. Combin. Theory Ser. B 92 (2004), 325–357.

    MathSciNet  Article  Google Scholar 

  26. C. P. Rourke and B. J. Sanderson: Introduction to piecewise-linear topology, Springer Study Edition. Springer-Verlag, Berlin-New York, 1982. Reprint.

    MATH  Google Scholar 

  27. M. Schaefer: Hanani-Tutte and related results, in: Geometry—intuitive, discrete, and convex, volume 24 of Bolyai Soc. Math. Stud., 259–299. János Bolyai Math. Soc., Budapest, 2013.

    Google Scholar 

  28. A. Schrijver and B. Sevenster: The strong arnold property for 4-connected flat graphs, Linear Algebra and its Applications 522 (2017), 153–160.

    MathSciNet  Article  Google Scholar 

  29. Z. Stanić: Regular Graphs: A Spectral Approach, De Gruyter Series in Discrete Mathematics and Applications. De Gruyter, 2017.

  30. D. R. Stinson: Combinatorial Designs: Construction and Analysis, Springer-Verlag New York, 2004.

    MATH  Google Scholar 

  31. M. Tait: The Colin de Verdière parameter, excluded minors, and the spectral radius, Journal of Combinatorial Theory, Series A 166 (2019), 42–58.

    MathSciNet  Article  Google Scholar 

  32. M. Tancer and D. Tonkonog: Nerves of good covers are algorithmically unrecognizable, SIAM J. Comput. 42 (2013), 1697–1719.

    MathSciNet  Article  Google Scholar 

  33. R. Thomas: Recent excluded minor theorems for graphs, in: In surveys in combinatorics, 1999, Ed. by J. D. Lamb and D. A. Preece. London Mathematical Society Lecture Note Series. Cambridge University Press, 1999, 201–222.

  34. G. M. Ziegler: Lectures on polytopes, volume 152 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.

    Google Scholar 

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We would like to thank Radek Husek and Robert Šámal for general discussions on the parameter We would also like to thank Arnaud de Mesmay for pointing us to the paper [12] and Rose McCarty for pointing us to [6].

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Correspondence to Vojtěch Kaluža.

Additional information

V. K. gratefully acknowledges the support of Austrian Science Fund (FWF): P 30902-N35. This work was done mostly while he was employed at the University of Innsbruck. During the early stage of this research, V. K. was partially supported by Charles University project GAUK 926416.

M. T. is supported by the grant no. 19-04113Y of the Czech Science Foundation (GAČR) and partially supported by Charles University project UNCE/SCI/004.

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Kaluža, V., Tancer, M. Even Maps, the Colin de Verdière Number and Representations of Graphs. Combinatorica (2022).

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Mathematics Subject Classification (2010)

  • 05C50
  • 57Q35
  • 05C10
  • 05C62