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

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Abstract

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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Acknowledgment

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].

Author information

Authors and Affiliations

  1. Institute of Science and Technology, Klosterneuburg, Austria

    Vojtěch Kaluža

  2. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Martin Tancer

Authors
  1. Vojtěch Kaluža
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  2. Martin Tancer
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Corresponding author

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 42 (Suppl 2), 1317–1345 (2022). https://doi.org/10.1007/s00493-021-4443-7

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  • DOI: https://doi.org/10.1007/s00493-021-4443-7

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