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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%2F22%3A10476095" target="_blank" >RIV/00216208:11320/22:10476095 - isvavai.cz</a>

  • Result on the web

    <a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=n1-X1aHe-f" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=n1-X1aHe-f</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1007/s00493-021-4443-7" target="_blank" >10.1007/s00493-021-4443-7</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 Colin de Verdiere graph parameter mu for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved mu(G) &lt;= sigma(G) + 2 and conjectured sigma(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 sigma(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 at much smaller values, namely, we exhibit a graph H for which mu(H) &gt;= 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

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • 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

    2022

  • Confidentiality

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

Data specific for result type

  • Name of the periodical

    Combinatorica

  • ISSN

    0209-9683

  • e-ISSN

    1439-6912

  • Volume of the periodical

    42

  • Issue of the periodical within the volume

    SUPPL 2 / Supplement 2

  • Country of publishing house

    DE - GERMANY

  • Number of pages

    29

  • Pages from-to

    1317-1345

  • UT code for WoS article

    000798210100003

  • EID of the result in the Scopus database

    2-s2.0-85130418329