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) <= sigma(G) + 2 and conjectured mu(G) <= 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) <= 18 and sigma(G) >= 20. We show that the gap appears on much smaller values, namely, we exhibit a graph H for which mu(H) <= 7 and sigma(H) >= 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) >= q(2).
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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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'20)
ISBN
978-1-61197-599-4
ISSN
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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