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Even maps, the Colin de Verdiere number and representations of graphs

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10421445" target="_blank" >RIV/00216208:11320/20:10421445 - isvavai.cz</a>

  • Result on the web

    <a href="https://doi.org/10.1137/1.9781611975994.161" target="_blank" >https://doi.org/10.1137/1.9781611975994.161</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1137/1.9781611975994.161" target="_blank" >10.1137/1.9781611975994.161</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Even maps, the Colin de Verdiere number and representations of graphs

  • Original language description

    Van der Holst and Pendavingh introduced a graph parameter sigma, which coincides with the more famous Cohn de Verdiere graph parameter mu for small values. However, the definition of sigma is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured mu(G) &lt;= sigma(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on mu(G) which is, in general, tight. Equality between mu and sigma does not hold in general as van der Holst and Pendavingh showed that there is a graph G with mu(G) &lt;= 18 and sigma(G) &gt;= 20. We show that the gap appears on much smaller values, namely, we exhibit a graph H for which mu(H) &lt;= 7 and sigma(H) &gt;= 8. We also prove that, in general, the gap can be large: The incidence graphs H-q of finite projective planes of order q satisfy mu(H-q) is an element of O(q(3/2)) and sigma(H-q) &gt;= q(2).

  • Czech name

  • Czech description

Classification

  • Type

    D - Article in proceedings

  • CEP classification

  • OECD FORD branch

    10101 - Pure mathematics

Result continuities

  • Project

    <a href="/en/project/GJ19-04113Y" target="_blank" >GJ19-04113Y: Advanced tools in combinatorics, topology and related areas</a><br>

  • Continuities

    P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Others

  • Publication year

    2020

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Article name in the collection

    PROCEEDINGS OF THE THIRTY-FIRST ANNUAL ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA&apos;20)

  • ISBN

    978-1-61197-599-4

  • ISSN

  • e-ISSN

  • Number of pages

    16

  • Pages from-to

    2642-2657

  • Publisher name

    ASSOC COMPUTING MACHINERY

  • Place of publication

    NEW YORK

  • Event location

    Salt Lake City

  • Event date

    Jan 5, 2020

  • Type of event by nationality

    WRD - Celosvětová akce

  • UT code for WoS article

    000554408102042